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LOGIC 

PART II 



CAMBRIDGE UNIVERSITY PRESS 

C. F. CLAY, MANAGER 
LONDON : FETTER LANE, B.C. 4 



NEW YORK : THE MACMILLAN CO. 

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TORONTO : THE MACMILLAN CO. OF 

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LOGIC 

PART II / 

DEMONSTRATIVE INFERENCE: 
DEDUCTIVE AND INDUCTIVE 



BY 



wfi5 JOHNSON, M.A. 

FELLOW OF KING S COLLEGE, CAMBRIDGE, 

SIDGWICK LECTURER IN MORAL SCIENCE IN THE 

UNIVERSITY OF CAMBRIDGE 



CAMBRIDGE 

AT THE UNIVERSITY PRESS 
1922 



Be 
nl 



CONTENTS 



INTRODUCTION 

PAGE 

i. Application of the term substantive ... xi 

2. Application of the term adjective . . . . . . . xii 

3. Terms substantive and adjective contrasted with particular and 

universal ........... xiii 

4. Epistemic character of assertive tie ....... xiv 

5. The given presented under certain determinables .... xiv 

6. The paradox of implication ........ xv 

7. Defence of Mill s analysis of the syllogism ...... xvii 



CHAPTER I 
INFERENCE IN GENERAL 

i. Implication denned as potential inference ...... i 

2. Inferences involved in the processes of perception and association . i 
3. Constitutive and epistemic conditions for valid inference. Examination 

of the paradox of inference ........ 7 

4. The Applicative and Implicative principles of inference . . . 10 
5. Joint employment of these principles in the syllogism . . .11 

6. Distinction between applicational and implicational universals. The 

structural proposition redundant as minor premiss . . . .12 
7. Definition of a logical category in terms of adjectival determinables . 15 
8. Analysis of the syllogism in terms of assigned determinables. Further 

illustrations of applicational universals . . . . . .17 

9. How identity may be said to be involved in every proposition . . 10 
10. The formal principle of inference to be considered redundant as major 

premiss. Illustrations from syllogism, induction, and mathematical 

equality ............ 20 

11. Criticism of the alleged subordination of induction under the syllogistic 
<x principle ............ 24 



vi CONTENTS 

CHAPTER II 

THE RELATIONS OF SUB-ORDINATION AND CO-ORDINA 
TION AMONGST PROPOSITIONS OF DIFFERENT TYPES 



i. The Counter-applicative and Counter-implicative principles required 
for the establishment of the axioms of Logic and Mathematics . . 

i. Explanation of the Counter-applicative principle .... 

3. Explanation of the Counter-implicative principle .... 

4. Significance of the two inverse principles in the philosophy of thought 

5. Scheme of super-ordination, sub-ordination and co-ordination amongst 
propositions 

6. Further elucidation of the scheme ....... 



CHAPTER III 
SYMBOLISM AND FUNCTIONS 







i. The value of symbolism. Illustrative and shorthand symbols. Classifi 
cation of formal constants. Their distinction from material constants . 41 
2. The nature of the intelligence required in the construction of a symbolic 

system .... ........ 44 

3. The range of variation of illustrative symbols restricted within some 
logical category. Combinations of such symbols further to be inter 
preted as belonging to an understood logical category. Illustrations of 
intelligence required in working a symbolic system .... 46 

4. Explanation of the term * function, and of the * variants for a function 48 
5. Distinction between functions for which all the material constituents 
are variable, and those for which only some are variable. Illustrations 
from logic and arithmetic ......... 50 

6. The various kinds of elements of form in a construct . . . . 53 

7. Conjunctional and predicational functions . . . . . -55 

8. Connected and unconnected sub-constructs . . . . -57 

9. The use of apparent variables in symbolism for the representation of the 
distributives every and some. Distinction between apparent variables 
and class-names .......... 58 

i 10. Discussion of compound symbols which do and which do not represent 

genuine constructs 61 

in. Illustrations of genuine and fictitious constructs 64 

i 12. Criticism of Mr Russell s view of the relation between prepositional 

functions and the functions of mathematics 66 

113. Explanation of the notion of a descriptive function .... 69 

14. Further criticism of Mr Russell s account of propositional functions . 71 

115. Functions of two or more variants ....... 73 



CONTENTS vii 

CHAPTER IV 
THE CATEGORICAL SYLLOGISM 

PAGE 

I. Technical terminology of syllogism ....... 76 

2. Dubious propositions to illustrate syllogism . . . . -77 

3. Relation of syllogism to antilogism . . . . . . 78 

4. Dicta for the first three figures derived from a single antilogistic dictum, 

showing the normal functioning of each figure ..... 79 

5. Illustration of philosophical arguments expressed in syllogistic form . 81 
6. Re-formulation of the dicta for syllogisms in which all the propositions 

are general ........... 83 

7. The propositions of restricted and unrestricted form in each figure . 84 

8. Special rules and valid moods for the first three figures ... 85 

9. Special rules and valid moods for the fourth figure .... 87 

} 10. Justification for the inclusion of the fourth figure in logical doctrine . 88 

j ii. Proof of the rules necessary for rejecting invalid syllogisms. . . 89 

5 12. Summary of above rules; and table of moods unrejected by the rules 

of quality ............ 92 

513. Rules and tables of unrejected moods for each figure .... 93 
} 14. Combination of the direct and indirect methods of establishing the valid 

moods of syllogism .......... 96 

115. Diagram representing the valid moods of syllogism .... 97 

\ 16. The Sorites 97 

}i7. Reduction of irregularly formulated arguments to syllogistic form . 98 

| 18. Enthymemes ........... 100 

119. Importance of syllogism . .102 



CHAPTER V 

FUNCTIONAL EXTENSION OF THE SYLLOGISM 

i. Deduction goes beyond mere subsumptive inference, when the major 

premiss assumes the form of a functional equation. Examples . . 103 
2. A functional equation is a universal proposition of the second order, the 

functional formula constituting a Law of Co-variation . . . . 105 

3. The solutions of mathematical equations which yield single-valued func 
tions correspond to the reversibility of cause and effect . . .106 
4. Significance of the number of variables entering into a functional formula 108 
5. Example of a body falling in vacua . . . . . . .no 

6. The logical characteristics of connectional equations illustrated by thermal 

and economic equilibria . . . . . . . . .in 

7. The method of Residues is based on reversibility and is purely deductive 1 16 
8. Reasons why the above method has been falsely termed inductive . 119 
9. Separation of the subsumptive from the functional elements in these 

extensions of syllogism . . . . ... . .120 



viii CONTENTS 

CHAPTER VI 
FUNCTIONAL DEDUCTION 

PAGE 

i. In the deduction of mathematical and logical formulae, new theorems 
are established for the different species of a genus, which do not hold 

for the genus 123 

2. Explanation of the Aristotelean tSiov . . . . . . .125 

3. In functional deduction, the equational formulae are non-limiting. 

Elementary examples . . . . . . . . .126 

4. The range of universality of a functional formula varies with the number 
of independent variables involved. Employment of brackets. Impor 
tance of distinguishing between connected and disconnected compounds 128 
5. The functional nature of the formulae of algebra accounts for the possi 
bility of deducing new and even wider formulae from previously estab 
lished and narrower formulae, the Applicative Principle alone being 
employed . . . . . . . . . . . .130 

6. Mathematical Induction . . . . . . . . 133 

7. The logic of mathematics and the mathematics of logic . . 135 

8. Distinction between premathematical and mathematical logic . .138 
9. Formal operators and formal relations represented by shorthand and 
not variable symbols. Classification of the main formal relations ac 
cording to their properties . . . . . . . . .141 

10. The material variables of mathematical and logical symbolisation receive 
specific values only in concrete science . . . . . .144 

11. Discussion of the Principle of Abstraction ...... 145 

12. The specific kinds of magnitude are not determinates of the single de- 
terminable Magnitude, but are incomparable . . . . .150 

13. The logical symbolic calculus establishes formulae of implication which 
are to be contrasted with the principles of inference employed in the 
procedure of building up the calculus . . . . . . .151 



CHAPTER VII 

THE DIFFERENT KINDS OF MAGNITUDE 

i. The terms greater and less predicated of magnitude, larger and 

smaller of that which has magnitude . . . . . 153 

2. Integral number as predicable of classes or enumerations . . .154 
3. Psychological exposition of counting . . . . . . -155 

4. Logical principles underlying counting . . . . . 158 

5. One-one correlations for finite integers 160 

6. Definition of extensive magnitude . . . . . . .161 

7. Adjectival stretches compared with substantival . . . . .163 

8. Comparison between extensive and extensional wholes . . . 166 
9. Discussion of distensive magnitudes . . . . . . .168 

10. Intensive magnitude . . . . . . . . . .172 

^n. Fundamental distinction between distensive and intensive magnitudes. 173 



CONTENTS ix 

PAGE 

12. The problem of equality of extensive wholes . . . . J 74 

13. Conterminus spatial and temporal wholes to be considered equal, quali 
tative stretches only comparable by causes or effects . . 175 
14. Complex magnitudes derived by combination of simplex . . .180 

15. The theory of algebraical dimensions 185 

1 6. The special case in which dividend and divisor are quantities of the 

same kind l8 6 

17. Summary of the above treatment of magnitude . . . . .187 



CHAPTER VIII 

INTUITIVE INDUCTION 

t. The general antithesis between induction and deduction . . .189 

2. The problem of abstraction 190 

3. The principle of abstractive or intuitive induction . . . .191 

4. Experiential and formal types of intuitive induction . . . .192 

5. Intuitive induction involved in introspective and ethical judgments . 193 

6. Intuitive inductions upon sense-data and elementary algebraical and 

logical relations . . . . . . . T 94 

7. Educational importance of intuitive induction . . . . .196 

CHAPTER IX 

SUMMARY INCLUDING GEOMETRICAL INDUCTION 

i. Summary induction reduced to first figure syllogism . . . 197 
2. Summary induction as establishing the premiss for induction proper. 

Criticism of Mill s and Whewell s views . . . . . .198 

3. Summary induction involved in geometrical proofs .... 200 

4. Explanation of the above process ....... 201 

5. Function of the figure in geometrical proofs ..... 203 

6. Abuse of the figure in geometrical proofs . . . . . . 205 

7. Criticism of Mill s parity of reasoning 208 

CHAPTER X 

DEMONSTRATIVE INDUCTION 

i. Demonstrative induction uses a composite along with an instantial 

premiss .... ......... 210 

2. Illustrations of demonstrative arguments leading up to demonstrative 

induction . . . . . . . . . . . .210 

3. Conclusions reached by the conjunction of an alternative with a dis 
junctive premiss . . - . . . . . . .214 



x CONTENTS 

PAGE 

4. The formula of direct universalisation . . . . . .215 

5. Scientific illustration of the above . . . . . . .216 

6. Proposed modification of Mill s exposition of the methods of induction 217 
7. The major premiss for demonstrative induction as an expression of the 
dependence in the variations of one phenomenal character upon those 
of others . . . . . . . . . . . .218 

8. The four figures of demonstrative induction . . . . .221 

9. Figure of Difference .......... 222 

10. Figure of Agreement . ......... 223 

11. Figure of Composition . . . . . . . . .224 

12. Figure of Resolution . ......... 226 

13. The Antilogism of Demonstrative Induction ..... 226 

14. Illustration of the Figure of Difference 228 

15. Illustration of the Figure of Agreement . . . . . . 231 

1 6. Principle for dealing with cases in which a number both of cause-factors 

and effect-factors are considered, with a symbolic example . . . 232 
17. Modification of symbolic notation in the figures where different cause- 
factors represent determinates under the same determinable . . 234 
1 8. The striking distinction between the two last and the two first figures . 235 
19. Explanation of the distinction between composition and combination 

of cause-factors ........... 235 

20. Illustrations of the figures of Composition and Resolution . . .237 



CHAPTER XI 

THE FUNCTIONAL EXTENSION OF DEMONSTRATIVE 
INDUCTION 

i. The major premiss for Demonstrative Induction must have been estab 
lished by Problematic Induction ....... 240 

2. Contrast between my exposition and Mill s . . . . .241 

3. The different uses of the term hypothesis in logic . . . .242 

4. Jevons s confusion between the notions ^problematic and hypo 
thetical ............ 244 

5. The establishment of a functional formula for the figures of Difference 

and of Composition 246 

6. The criteria of simplicity and analogy for selection of the functional 

formula ............ 249 

7. A comparison of these criteria with similar criteria proposed by 

Whewell and Mill 251 

8. Technical mathematical methods for determining the most probable 

formula ..... ....... 252 

INDEX 254 



INTRODUCTION TO PART II 

i. BEFORE introducing the topics to be examined 
in Part II, I propose to recapitulate the substance of 
Part I, and in so doing to bring into connection with 
one another certain problems which were there treated 
in different chapters. I hope thus to lay different em 
phasis upon some of the theories that have been main 
tained, and to remove any possible misunderstandings 
where the treatment was unavoidably condensed. 

In my analysis of the proposition I have distinguished 
the natures of substantive and adjective in a form in 
tended to accord in essentials with the doctrine of the 
large majority of logicians, and as far as my terminology 
is new its novelty consists in giving wider scope to each 
of these two fundamental terms. Prima facie it might 
be supposed that the connection of substantive with 
adjective in the construction of a proposition is tanta 
mount to the metaphysical notions of substance and 
inherence. But my notion of substantive is intended 
to include, besides the metaphysical notion of substance 
so far as this can be philosophically justified the no 
tion of occurrences or events to which some philosophers 
of the present day wish to restrict the realm of reality. 
Thus by a substantive proper I mean an existent; and 
the category of the existent is divided into the two 
subcategories: what continues to exist, or the continuant; 
and what ceases to exist, or the occurrent, every occur- 
rent being referable to a continuant. To exist is to be 



xii INTRODUCTION 

in temporal or spatio-temporal relations to other exis- 
tents; and these relations between existents are the 
fundamentally external relations. A substantive proper 
cannot characterise, but is necessarily characterised ; on 
the other hand, entities belonging to any category 
whatever (substantive proper, adjective, proposition, 
etc.) may be characterised by adjectives or relations 
belonging to a special adjectival sub-category corre 
sponding, in each case, to the category of the object 
which it characterises. Entities, other than substantives 
proper, of which appropriate adjectives can be predi 
cated, function as quasi-substantives. 

2. The term adjective, in my application, covers 
a wider range than usual, for it is essential to my system 
that it should include relations. There are two distinct 
points of view from which the treatment of a relation 
as of the same logical nature as an adjective may be 
defended. In the first place the complete predicate in 
a relational proposition is, in my view, relatively to the 
subject of such proposition, equivalent to an adjective 
in the ordinary sense. For example, in the proposition, 
He is afraid of ghosts, the relational component is ex 
pressed by the phrase afraid of ; but the complete 
predicate afraid of ghosts (which includes this relation) 
has all the logical properties of an ordinary adjective, 
so that for logical purposes there is no fundamental dis 
tinction between such a relational predicate and an irra 
tional predicate. In the second place, if the relational 
component in such a proposition is separated, I hold that 
it can be treated as an adjective predicated of the sub 
stantive-couple he and ghosts . In other words, a rela 
tion cannot be identified with a class of couples, i.e. be 



INTRODUCTION xiii 

conceived extensionally ; but must be understood to 
characterise couples, i.e. be conceived intensionally. It 
seems to me to raise no controvertible problem thus to 
include relations under the wide genus adjectives. It is 
compatible, for example, with almost the whole of Mr 
Russell s treatment of the proposition in his Principles of 
Mathematics-, and, without necessarily entering into the 
controvertible issues that emerge in such philosophical 
discussions, I hold that some preliminary account of 
relations is required even in elementary logic. 

3. My distinction between substantive and adjec 
tive is roughly equivalent to the more popular philoso 
phical antithesis between particular and universal; the 
notions, however, clo not "elcactty coincide. Thus I 
understand the philosophical term particular not to apply 
to quasi-substantives, but to be restricted to substantives 
proper, i.e. existents, or even more narrowly to occur- 
rents. On the other hand, I find a fairly unanimous 
opinion in favour of calling an adjective predicated of 
a particular subject, a particular the name universal 
being confined to the abstract conception of the adjec 
tive. Thus red or redness, abstracted from any specific 
judgment, is held to be universal; but the redness, 
manifested in a particular object of perception, to be 
itself particular. Furthermore, qua particular, the ad 
jective is said to be an existent, apparently in the same 
sense as the object presented to perception is an exis 
tent. To me it is difficult to argue this matter because, 
while acknowledging that an adjective may be called a 
universal, I regard it not as a mere abstraction, but as 
a factor in the real ; and hence, in holding that the ob 
jectively real is properly construed into an adjective 



xiv INTRODUCTION 

characterising a substantive, the antithesis between the 
particular and the universal (i.e. in my terminology 
between the substantive and the adjective) does not 
involve separation within the real, but solely a separation 
for thought, in the sense that the conception of the 
substantive apart from the adjective, as well as the 
conception of the adjective apart from the substantive, 
equally entail abstraction. 

4. Again, taking the whole proposition constituted 
by the connecting of substantive with adjective, I have 
maintained that in a virtually similar sense the proposi 
tion is to be conceived as abstract. But, whereas the 
characterising tie may be called constitutive in its func 
tion of connecting substantive with adjective to con 
struct the proposition, I have spoken of the assertive 
tie as epistemic, in the sense that it connects the thinker 
with the proposition in constituting the unity which may 
be called an act of judgment or of assertion. When, 
however, this act of assertion becomes in its turn an 
object of thought, it is conceived under the category of 
the existent; for such an act has temporal relations to 
other existents, and is necessarily referrible to a thinker 
conceived as a continuant. Though, relatively to the 
primary proposition, the assertive tie must be conceived 
as epistemic ; yet, relatively to the secondary proposition 
which predicates of the primary that it has been asserted 
by A, the assertive tie functions constitutively. 

5. In view of a certain logical condition presup 
posed throughout this Part of my work, I wish to re 
mind the reader of that aspect of my analysis of the 
proposition, according to which I regard the subject as 
that which is given to be determinately characterised 



INTRODUCTION xv 

by thought. Now I hold that for a subject to be 
characterised by some adjectival determinate, it must 
first have been presented as characterised by the corre 
sponding adjectival determinable. The fact that what 
is given is characterised by an adjectival determinable 
is constitutive ; but the fact that it is presented as thus 
characterised is epistemic. Thus, for a surface to be 
characterised as red or as square, it must first have 
been constructed in thought as being the kind of thing 
that has colour or shape ; for an experience to be 
characterised as pleasant or unpleasant, it must first 
have been constructed in thought as the kind of thing 
that has hedonic tone. Actually what is given, is to be 
determined with respect to a conjunction of several 
specific aspects or determinables ; and these determine 
the category to which the given belongs. For example, 
on the dualistic view of reality, the physical has to be 
determined under spatio-temporal determinables, and 
the psychical under the determinable consciousness or 
experience. If the same being can be characterised as 
two-legged and as rational, he must be put into the 
category of the physico-psychical. 

6. The passage from topics treated in Part I to 
those in Part II, is equivalent to the step from implica 
tion to inference. The term inference, as introduced in 
Part I, did not require technical definition or analysis, 
as it was sufficiently well understood without explana 
tion. It was, however, necessary in Chapter III to in 
dicate in outline one technical difficulty connected with 
the paradox of implication ; and there I first hinted, 
what will be comprehensively discussed in the first 
chapter of this Part, that implication is best conceived 



xvi INTRODUCTION 

as potential inference. While for elementary purposes 
implication and inference may be regarded as practically 
equivalent, it was pointed out in Chapter III that there 
is*nevertheless one type of limiting condition upon which 
depends the possibility of using the relation of implica 
tion for the purposes of inference. Thus reference to 
the specific problem of thejaaradox of implication was 

V TT A * I I III i 1 1 ,_. ,... ...a 

jf unavoidable in Part I, inasmuch as a comprehensive 
account of symbolic and mechanical processes necessarily 
included reference to all possible limiting cases; but, 
apart from such a purely abstract treatment, no special 
logical importance was attached to the paradox. The 
limiting case referred to was that of the permissible em 
ployment of the compound proposition If/ then ^, in the 
unusual circumstance where knowledge of the truth or 
the falsity of/ or of q was already present when the com 
pound proposition was asserted. This limiting case will 
not recur in the more important developments of infer 
ence that will be treated in the present part of my logic. 
It might have conduced to greater clearness if, in 
Chapters III and IV, I had distinguished when using 
the phrase imp lie ative proposition between the primary 
and secondary interpretations of this form of proposi 
tion. Thus, when the compound proposition If/ then 
q is rendered, as Mr Russell proposes, in the form 
Either not-/ or.^, the compound is being treated as a 
primary proposition of the same type as its components 
/ and q. When on the other hand we substitute for If 
/ then q the phrase / implies q] or preferably / would 
imply .q* the proposition is no longer primary, inasmuch 
as it predicates about the proposition q the adjective 
implied by/ which renders the compound a secondary 



INTRODUCTION xvii 

proposition, in the sense explained in Chapter IV 1 . Now 
whichever of these two interpretations is adopted, the 
inference which is legitimate under certain limiting con 
ditions is the same. Thus given the compound Either 
not-/ or q conjoined with the assertion of *// we could 
infer q\ just as given p implies q conjoined with the 
assertion of ^vjv^jnjer^. 1 It is for this reason that 
the two interpretations have become merged into one 
in the ordinary symbolic treatment of compound pro 
positions; and in normal cases no distinction is made 
in regard to the possibility of using the primary or 
secondary interpretation for purposes of inference. The 
normal case, however, presupposes that p and q are 
entertained hypothetically; when this does not obtain, 
the danger of petitio principii enters. The problem in 
Part I was only a very special and technical case in 
which this fallacy has to be guarded against ; in Part II, 
it will be dealt with in its more concrete and philoso 
phically important applications. 

7. The mention of this fallacy immediately sug 
gests Mill s treatment of the functions and value of the 
syllogism; but, before discussing his views, I propose 
to consider what his main purpose was in tackling the 
charge of petitio principii that had been brought against 
the whole of formal argument, including in particular 
the syllogism. In the first section of his chapter, Mill 
refers to two opposed classes of philosophers the one 
of whom regarded syllogism as the universal type of all 
logical reasoning, the other of whom regarded syllogism 

1 The interpretation of the implicative form */ implies q as 
secondary is developed in Chapter III, 9, where the modal adjectives 
necessary, possible, impossible, are introduced. 



xviii INTRODUCTION 

as useless on the ground that all such forms of inference 
involve petitio principii. He then proceeds: I believe 
both these opinions to be fundamentally erroneous/ and 
this would seem to imply that he proposed to relieve 
the syllogism from the charge. I believe, however, that 
all logicians who have referred to Mill s theory a 
group which includes almost everyone who has written 
on the subject since his time have assumed that the 
purport of the chapter was to maintain the charge of 
petitio principii, an interpretation which his opening 
reference to previous logicians would certainly not seem 
to bear. His subsequent discussion of the subject is, 
verbally at least, undoubtedly confusing, if not self-con 
tradictory ; but my personal attitude is that, whatever 
may have been Mill s general purpose, it is from his own 
exposition that I, in common with almost all his con 
temporaries, have been led to discover the principle 
according to which the syllogism can be relieved from 
the incubus to which it had been subject since the time 
of Aristotle. In my view, therefore, Mill s account of 
the philosophical character of the syllogism is incon 
trovertible ; I would only ask readers to disregard from 
the outset any passage in his chapter in which he 
appears to be contending for the annihilation of the 
syllogism as expressive of any actual mode of inference. 
Briefly his position may be thus epitomised. Taking 
a typical syllogism with the familiar major All men 
are mortal, he substituted for Socrates or Plato the 
minor term the Duke of Wellington who was then 
living. He then maintained that, going behind the 
syllogism, certain instantial evidence is required for es 
tablishing the major; and furthermore that the validity 



INTRODUCTION xix 

of the conclusion that the Duke of Wellington would 
die depends ultimately on this instantial evidence. The 
interpolation of the universal major All men will die 
has undoubted value, to which Mill on the whole did 
justice; but he pointed out that the formulation of this 
universal adds nothing to the positive or factual data 
upon which the conclusion depends. It follows from 
his exposition that a syllogism whose major is admittedly 
established by induction from instances can be relieved 
from the reproach of begging the question or circularity 
if, and only if, the minor term is not included in the 
ultimate evidential data. The Duke of Wellington being 
still living could not have formed part of the evidence 
upon which the universal major depended. It was there 
fore part of Mill s logical standpoint to maintain that 
there were principles of induction by which, from a 
limited number of instances, a universal going beyond 
these could be logically justified. This contention may 
be said to confer constitutive validity upon the inductive 
process. It is directly associated with the further con 
sideration that an instance, not previously examined, may 
be adduced to serve as minor premiss for a syllogism, 
and that such an instance will always preclude circularity 
in the formal process. Now the charge of circularity or 
petitio principii is epistemic; and the whole of Mill s 
argument may therefore be summed up in the statement 
that the epistemic validity of syllogism and the consti 
tutive validity of induction, both of which had been dis 
puted by earlier logicians, stand or fall together. 

In order to prevent misapprehension in regard to 
Mill s view of the syllogism, it must be pointed out that 
he virtually limited the topic of his chapter to cases in 



xx INTRODUCTION 

which the major premiss would be admitted by all 
logicians to have been established by means of induction 
in the ordinary sense, i.e. by the simple enumeration of 
instances; although many of them would have contended 
that such instantial evidence was not by itself sufficient. 
Thus all those cases in which the major was otherwise 
established, such as those based on authority, intuition 
or demonstration, do not fall within the scope of Mill s 
solution. Unfortunately all the commentators of Mill 
have confused his view that universals cannot be in 
tuitively but only empirically established, with his spe 
cific contention in Chapter IV. I admit that he himself 
is largely responsible for this confusion, and therefore, 
while supporting his view on the functions of the syl 
logism, I must deliberately express my opposition to 
his doctrine that universals can only ultimately be estab 
lished empirically, and limit my defence to his analysis 
of those syllogisms in which it is acknowledged that the 
major is thus established. Even here his doctrine that 
all inference is from particulars to particulars is open to 
fundamental criticism ; and, in my treatment of the 
principles of inductive inference which will be developed 
in Part III, I shall substitute an analysis which will 
take account of such objections as have been rightly 
urged against Mill s exposition. 

[NOTE. There are two cases in which the technical terminology 
employed in Part II differs from that in Part I. (i) The phrase/r/w*- 
tive proposition, in Part I, is to be understood psychologically; in 
Part II, logically as equivalent to axiom. (2) Counter-implicative, in 
Parti, applies to the form of a compound proposition; in Part II, to 
a principle of inference.] 



X 



CHAPTER I 

INFERENCE IN GENERAL 

i. INFERENCE is a mental process which, as s 
has to be contrasted with implication. The connection 
between the mental act of inference and the rdatipn 
of implication is analogous to that between assertion and 
the proposition. Just as a proposition^ js what is poten 
tially assertible, so the relation^of JmgljrgjJQn between 
two propositions is an essential condition for the possi 
bility of inferring one from the other; and, as it is 
impossible to define a proposition ultimately except in 
terms of the notion of asserting, so the rd^^g^^Jm^- 
plication can only be. defined in "terms of inference./ 
This consideration explains the importance which I , 
attach to the recognition of the mental attitude involved f 
in inference and assertion ; afterwhich the strictly logical 
question as to the distinction between valid and invalid 
inference can be discussed. To distinguish the formula 
of implication from that of inference, the former may 
be symbolised IfjJ then qj and the latter p therefore 
ql where the symbol q stands for the conclusion andp 
for the premiss or conjunction of premisses. 

The proposition or propositions from which an in 
ference is made being called premisses, and the pro 
position inferred being called the conclusion, it is 
commonly supposed that the premisses are the pro 
positions first presented in thought, and that the transi 
tion from these to the thought of the conclusion is the 

j. L. ii i 



2 CHAPTER I 

last step in the process. But in fact the reverse is 
usually the case; that is to say, we first entertain in 
thought the proposition that is technically called the 
conclusion, and then proceed to seek for other pro 
positions which would justify us in assgd".ingL it. The 
conclusion may, on the one hand, first present itself to 
us as potentially assertible, in which case the mental 
process of inference consists in transforming what was 
potentially assertible into a.proposition actually asserted. 
On the other hand, we may have already satisfied 
ourselves that the conclusion can be validly asserted 
apart from the particular inferential process, in which 
case we may yet seek for other propositions which, 
functioning as premisses, would give an independent or 
additional justification for our original assertion. J& 
e -^giy__g^se, the process of inference involves__three jis- 
tinct assertions : first the assertion of ^, next the asser- 



thirdly the assertion that p would imply 
2/ It must be noted that { p would imply q} which is 
the proper equivalent of if/ then q] is the more correct 
expression for the relation of implication, and not l p 
implies q which rather expresses the completed in 
ference. TscthaMiference canntbe defined 




in terms of implication, but that implication must be \ 

defined in terms of inference, namely as equivalent to 
potential inference. T^raspnfiT^^ 

merely passing from thejtssertjnn of the premiss to_t:he 
assertionr-gf the conclusion, but we.^are also implicitly 
asserting that the assertion of the premiss is used to 
justify the assertion of the conclusion. 

2. Some difficult problems, which are of special 
importance in psychology, arise in determining quite 



INFERENCE IN GENERAL 3 

precisely the range of those mental processes which 
may be called inference , in particular, how far asser 
tion or inference is involved in the processes of asso 
ciation and of perception. These difficulties have been 
aggravated rather than removed by the quite false 
antithesis which some logicians have drawn between 
logical and psychological inference. Every inference is 
a mental process, and therefore a proper topic for psycho 
logical analysis ; on the other hand, to infer is to think, 
and to think is virtually to adopt a logical attitude; for 
everyone who infers, who asserts, who thinks, intends 
to assert truly and to infer validly, and this is what con 
stitutes assertion or inference into a logical process. It 
is the concern of the science of logic, as contrasted with 
psychology, to criticise such assertions and inferences 
from the point of view of their validity or invalidity. 

Let us then consider certain mental processes in 
particular processes of association which have the 
semblance of inference. In the first place, there are 
many unmistakeable cases of association in which no 
inference whatever is even apparently involved. Any 
familiar illustration, either of contiguity or of similarity, 
will prove that association in itself does not entail in 
ference. If a cloudy sky raises memory-images of a 
storm, or leads to the mental rehearsal of a poem, or 
suggests the appearance of a slate roof, in none of these 
revivals by association is there involved anything in the 
remotest degree resembling inference. The case of con 
tiguity is that which is most commonly supposed to 
involve some sort of inference; but in this supposal there 
is a confusion between recollection and expectation. 
Our recollection of storms that we have experienced in 



I 2 



4 CHAPTER I 

the past is obviously distinct from our expectation that 
a storm is coming on in the immediate future. It is to 
this latter process of expectation, and not to the former 
process of recollection, that the term inference is more 
or less properly applied ; but even here we must make 
a careful psychological distinction. We may expect a 
storm when we notice the darkness of the sky, without 
at all having actually recalled past experiences of storms; 
in this case no inference is involved, since there has 
been only one assertion, namely, what would constitute 
the conclusion without any other assertion that would 
constitute a premiss. In order to speak properly of 
inference in such cases, the minimum required is the 
assertion that the sky is cloudy and that therefore there 
will be a storm. Here we have two explicit assertions, 
together with the inference involved in the word there 
fore. It is of course a subtle question for introspection 
as to whether this threefold assertion really takes place. 
This difficulty does not at all affect our definition of 
inference; it would only affect the question whether in 
any given case inference had actually occurred. It has 
been suggested that, where there has been nothing that 
logic could recognise as an inference, there has yet 
been inference in a psychological sense; but this con 
tention is absurd, since it is entirely upon psychological 
grounds that we have denied the existence of inference 
in such cases. 

Let us consider further the logical aspects of a 
genuine inference, following upon such a process of 
association as we have illustrated. The scientist may 
hold that the appearance of the sky is not such as to 
warrant the expectation of an on-coming storm. He 



INFERENCE IN GENERAL 5 

may, therefore, criticise the inference as invalid. Thus, 
assuming the actuality of the inference from the psycho 
logical point of view, it may yet be criticised as invalid 
from the logical point of view. So far we have taken 
the simplest case, where the single premiss The sky 
is cloudy is asserted. But, when an additional premiss 
such as In the past cloudy skies have been followed 
by storm is asserted, then the inference is further 
rationalised, since the two premisses taken together 
constitute a more complete ground for the conclusion 
than the single premiss. This additional premiss is 
technically known as a particular proposition. If the 
thinker is pressed to find still stronger logical warrant 
for his conclusion, he may assert that in all his expe 
riences cloudy skies have been followed by storm (a 
limited universal). The final stage of rationalisation is 
reached when the universal limited to all remembered 
cases is used as the ground for asserting the unlimited 
universal for all cases. But even now the critic may 
press for further justification. To pursue this topic 
would obviously require a complete treatment of induc 
tion, syllogism, etc., from the logical point of view. 
Enough has been said to show that, however inade 
quate may be the grounds offered in justification of a 
conclusion, this has no bearing upon the nature or upon 
the fact of inference as such, but only upon the criticism 
of it as valid or invalid. 

As in association, so also in perception, a psycho 
logical problem presents itself. There appear to be at 
least three questions in dispute regarding the nature of 
perception, which have close connection with logical 
analysis: First, how much is contained in the percept 



6 CHAPTER I 

besides the immediate sense experience? Secondly, 
does perception involve assertion? Thirdly, does it 
involve inference ? To illustrate the nature of the first 
problem, let us consider what is meant by the visual 
perception of a match-box. This is generally supposed 
to include the representation of its tactual qualities ; in 
which case, the content of the percept includes qualities 
other than those sensationally experienced. On the 
other hand, supposing that an object touched in the 
dark is recognised as a match-box, through the special 
character of the tactual sensations, would the represen 
tation of such visual qualities as distinguish a match-box 
from other objects be included in the tactual perception 
of it as a match-box? The same problem arises when 
we recognise a rumbling noise as indicating a cart in 
the road: i.e. should we say, in this case, that the 
auditory percept of the cart includes visual or other dis 
tinguishing characteristics of the cart not sensationally 
experienced? In my view it is inconsistent to include in 
the content of the visual percept tactual qualities not 
sensationally experienced, unless we also include in the 
content of a tactual or auditory percept visual or similar 
qualities not sensationally experienced 1 . 

This leads up to our second question, namely whether 
in such perceptions there is an assertion (a) predicating 
of the experienced sensation certain specific qualities; 
or an assertion (K) of having experienced in the past 
similar sensations simultaneously with the perception of 

1 In speaking here of the mental representation of qualities not 
sensationally experienced, .1 am putting entirely aside the very im 
portant psychological question as to whether such mental repre 
sentations are in the form of sense-imagery 7 or of ideas. 



INFERENCE IN GENERAL 7 

a certain object. Employing our previous illustration, 
we may first question whether the assertion There is 
a cart in the road following upon a particular auditory 
sensation, involves (a) the explicit characterisation 
of that sensation. Now if the specific character of the 
noise as a sensation merely caused a visual image which 
in its turn caused the assertion There is a cart in the 
road, then in the absence of assertion (a) there is no 
explicit inference. In order to become inference, the 
character operating (through association) as cause would 
have to be predicated (in a connective judgment) as 
ground. On the other hand, any experience that could 
be described as hearing a noise of a certain more or less 
determinate character would involve, in my opinion, 
besides assimilation, a judgment or assertion (a) expres 
sible in some such words as There is a rumbling noise. 
The further assertion that there is a cart in the road 
is accounted for (through association) by previous ex 
periences of hearing such a noise simultaneously with 
seeing a cart. Assuming that association operates by 
arousing memory-images of these previous experiences, 
it is only when by their vividness or obtrusiveness these 
memory-images give rise to a memory -judgment, that 
the assertion (b) occurs. We are now in a position to 
answer the third question as to the nature of perception ; 
for, if either the assertion of (a) alone or of (b) with (a) 
occurs along with the assertion that there is a cart in 
the road, then inference is involved; otherwise it is not. 
3. Passing from the psychological to the strictly 
logical problem, we have to consider in further detail 
the conditions jbr th^^^liditv of an inference symbolised 
as l p . . q> These conditions are twofold, and may be 




8 CHAPTER I 

conveniently distinguished in accordance with my termi 
nology as constitutive and epistemic. They may be 
briefly formulated as follows: 

Conditions for Validity of the Inference l p . . q 

Constitutive Conditions , (i) the proposition / and 
(ii) the proposition p would imply q 9 must both be true. 

Epistemic Conditions , (i) the asserting of l p 9 and 
(ii) the asserting of p would imply q* must both be 
permissible without reference to the asserting of q. 

It will be noted that the constitutive condition ex 
hibits the dependence of inferential validity upon a 
certain relation between the contents of premiss and of 
conclusion ; the epistemic condition, upon a certain 
relation between the asserting of the premiss and the 
asserting of the conclusion. Taking the constitutive 
| condition first, we observe that the distinction between 
inference and implication is sometimes expressed by 
calling implication hypothetical inference the mean 
ing of which is that, in the act of inference, the premiss 
must be categorically asserted ; while, in the relation of 
i implication, this premiss is put forward merely hypo- 
|thetically. This was anticipated above by rendering 
the relation of implication in the subjunctive mood 
(p would imply q) and the relation of inference in the 
indicative mood (p implies q}. 

Further to bring out the connection between the 
epistemic and the constitutive conditions, it must be 
pointed out that an odd confusion attaches to the use 
of the word imply in these problems. The almost 
universal application of the relation of implication in 
logic is as a relation between two propositions; but, in 
familiar language, the term imply is used as a relation 



INFERENCE IN GENERAL 

between two assertions. Consider for instance (a) 
asserting that there will be a thunderstorm would imply 
his having noticed the closeness of the atmosphere, and 
(b) the closeness of the atmosphere would imply that 
there will be a thunderstorm. The first of these relates 
two mental acts of the general nature of assertion, and 
is an instance of the asserting of q would imply having 
asserted/ ; the second is a relation between two pro 
positions, and is an instance of the proposition/ would 
imply the proposition q Comparing (a) with (6) we 
find that implicans and implicate have changed places. 
Indeed the sole reason why the asserting of the thunder 
storm was supposed to imply having asserted the close 
ness of the atmosphere was that, in the speaker s judg 
ment, the closeness of the atmosphere would imply that 
there will be a thunderstorm. 

Recognising, then, this double and sometimes am 
biguous use of the word imply, we may restate the 
first of the two epistemic conditions and the second of 
the two constitutive conditions for the validity of the 
inference p . . q as follows: 

Epistemic condition (i) : the asserting of the propo 
sition / should not have implied the asserting of the 
proposition $. 

Constitutive condition (ii) : the proposition / should 
imply the proposition q" 

The former is merely a condensed equivalent of our 
original formulation, viz. that the asserting of the pro 
position / must be permissible without reference to the 
asserting of the proposition q. 

Now the fact that there is this double use of the 
term imply accounts for the paradox long felt as 



io CHAPTER I 

regards the nature of inference: for it is urged that, in 

(order that an inference may be formally valid, it is 
required that the conclusion should be contained in the 
premiss or premisses; while, on the other hand, if there 
is any genuine advance in thought, the conclusion must 
not be contained in the premiss. This word contained 7 
is doubly ambiguous: for, in order to secure formal 
validity, the premisses regarded as propositions must 
imply the conclusion regarded as a proposition; but, in 
order that there shall be some real advance and not a 
mere petitio principii, it is required that the asserting 
of the premisses should not have implied the previous 
asserting of the conclusion. These two horns of the 
dilemma are exactly expressed in the constitutive and 
epistemic conditions above formulated. 

4. We shall now explain how the constitutive 
conditions for the validity of inference, which have been 
expressed in their most general form, are realised in 
familiar cases. The general constitutive condition p 
jwould imply q is y^r^//)Tsatisfied when some specific 
logical relation holds of^> to y; and^ it is upon such a 
relation that the formal truth of _the assertion that p 
would imply q" is based. There are two fundamental 
relations which will render the inference from p to q, 
not only valid, but formally valid ; and these relations 
will be expressed in formulae exhibiting what will be 
called the Applicative and the Implicative Principles 
of Inference. The former may be said to formulate what 

I is involved in the intelligent use of the word every ; 

I the latter what is involved in the intelligent use of 
the word if. 

In formulating the Applicative principle, we take p 



INFERENCE IN GENERAL n 

to stand for a proposition universal in form, and q for 
a singular proposition which predicates of some single 
case what is predicated universally in /. The Appli 
cative principle will then be formulated as follows: 

From a predication about every we may formally 
infer the same predication about any given. 

In formulating the Implicative principle, we take^> 
to stand for a compound proposition of the form *x and 
"x implies y" and q to stand for the simple proposition 
y. y The Implicative principle will then be formulated 
as follows: 

From the compound proposition *x and "x implies 
y" we may formally infer 4 jy. 

5. We find two different forms of proposition, one 
or other of which is used as a premiss in every formal 
inference; the distinction between which is funda 
mental, but has been a matter of much controversy 
among logicians. In familiar logic the two kinds of 
proposition to which I shall refer are known respec 
tively as universal and hypothetical. As an example of 
the former, take Every proposition can be subjected 
to logical criticism ; from this universal proposition we 
may directly infer That "matter exists" can be sub 
jected to logical criticism. This inference illustrates 
what I have called the Applicative Principle, and its 
premiss will be called an Applicational universal. Take 
next the example If this can swim it breathes, and it 
can swim ; from this conjunction of propositions we 
infer that it breathes ; here, the hypothetical premiss 
being in our terminology called implicative, the in 
ference in question illustrates the use of the Implica- 



12 CHAPTER I 

tive Principle. It is the combination of these two prin 
ciples that marks the advance made in passing from 
the most elementary forms of inference to the syllogism. 
For example: From Everything breathes if able to 
swim we can infer This breathes if able to swim, 
where the applicative principle only is employed. Con 
joining the conclusion thus obtained with the further 
premiss This can swim, we can infer this breathes/ 
where the implicative principle only is employed. In 
this analysis of the syllogism which involves the inter 
polation of an additional proposition, we have shown 
how the two principles of inference are successively 
employed. The ordinary formulation of the syllogism 
would read as follows: Everything that can swim 
breathes; this can swim; therefore this breathes. In 
place of the usual expression of the major premiss, I 
have substituted Everything breathes if able to swim/ 
in order to show how the major premiss prepares the 
way for the inferential employment successively of the 
applicative and of the implicative principles. 

6. Now the two propositions Every proposition 
can be subjected to logical criticism and everything 
that is able to swim breathes must be carefully con 
trasted. Both of them are universal in form; but in the 
latter the subject term contains an explicit characterising 
adjective, viz. able to swim. The presence of a charac 
terising adjective in the subject anticipates the occasion 
on which the question would arise whether this adjec 
tive is to be predicated of a given object. In the 
syllogism, completed as in the preceding section, the 
universal major premiss is combined with an affirmative 
minor premiss, where the adjective entertained cate- 



INFERENCE IN GENERAL 13 

gorically as predicate of the minor is the same as that 
which was entertained hypothetically as subject of the 
major. This double functioning of an adjective is the 
one fundamental characteristic of all syllogism ; where 
it will be found that one (or, in the fourth figure, every) 
term occurs once in the subject of a proposition, where 
it is entertained hypothetically, and again in the pre 
dicate of another proposition where it is entertained 
categorically. 

The essential distinction between the two contrasted 
universals (applicational and implicational) lies in the 
fact that an inference can be drawn from the former on 
the applicative principle alone, which dispenses with 
the minor premiss. We have to note the nature of the 
substantive that occurs in the applicational universal as 
distinguished from that which occurs in the implicational 
universal. The example already given contained pro 
position as the subject term, and a few other examples 
are necessary to establish the distinction in question. 
Every individual is self-identical, 5 therefore the author 
of the Republic is self-identical ; Every conjunction 
of predications is commutative, therefore the conjunc 
tion lightning before and thunder after is commutative ; 
Every adjective is a relatively determinate specifica 
tion of a relatively indeterminate adjective, therefore 
red is a relatively determinate specification of a rela 
tively indeterminate adjective. These illustrations 
could be endlessly multiplied, in which we directly 
apply a universal proposition to a certain given instance. 
In "such cases the implicative as well as the applicative 
principle would have been involved if it had been 
necessary or possible to interpolate, as an additional 



I 4 CHAPTER I 

datum, a categorical proposition requiring certification, 
to serve as minor premiss. Let us i turn to "our original 
illustration and examine what would have been involved 
if we had treated the inference as a syllogism ; it would 
have read as follows: Every proposition can be sub 
jected to logical criticism ; That matter exists is a 
proposition ; therefore That matter exists can be sub 
jected to logical criticism/ In this form, the substantive 
word proposition occurs as subject in the universal 
premiss, and as predicate in the singular premiss. What 
I have to maintain is that this introduction of a minor 
premiss is superfluous and even misleading. It should 
be observed that, in all the illustrations given above of 
the purely applicative principle, the subject-term in the 
universal premiss denotes a general category. It follows 
from this that the proposed statement That matter 
exists is a proposition is redundant as a premiss; for it 
is impossible for us to understand the meaning of the 
phrase matter exists except so far as we understand 
it to denote a proposition. In the same way, it would 
be impossible to understand the word red without 
understanding it to denote an adjective ; and so in all 
other cases of the pure employment of the applicative 
principle. In all these cases, the minor premiss which 
might be constructed is not a genuine proposition the 
truth of which could come up for consideration 
because the understanding of the subject-term of the 
minor demands a reference of it to the general category 
there predicated of it. This proposed minor premiss, 
therefore, is a peculiar kind of proposition which is not 
exactly what Mill calls verbal, but rather what Kant 
meant by analytic, and which I propose to call struc- 



INFERENCE IN GENERAL 15 

tural? All structural statements contain as their pre 
dicate some wide logical category, and their fundamental 
characteristic is that it is impossible to realise the 
meaning of the subject-term without implicitly con 
ceiving it under that category. The structural propo 
sition can hardly be called verbal, because it does not 
depend upon any arbitrary assignment of meaning to 
a word; this point being best illustrated by giving 
examples. For instance, taking as subject-term the 
author of the Republic, then The author of the 
Republic wrote something, would be verbal, while 
The author of the Republic is an individual, would 
be structural. In reality the subject of a verbal pro 
position, and the subject of a structural proposition are 
not the same; the one has for its subject the phrase the 
author of the Republic , and the other the object denoted 
by the phrase. This is the true and final principle for 
distinguishing a structural (as well as a genuinely real 
or synthetic statement) from a verbal statement. 

7. Since a category is expressed always by a 
general substantive name, the important question arises 
as to whether or how the name of a category such as 
existent or proposition is to be defined. Now the 
ordinary general substantive name is defined in terms 
of determinate adjectives which constitute its connota 
tion; but, so far as a category can be defined, it must 
be in terms of adjectival determinables\ e.g. an existent 
is what occupies some region of space or period of 
time: the determinates corresponding to which would 
be, occupying some specific region of space or period of 
time. Similarly, the category proposition could be 
defined by the adjectival determinable that to which 



16 CHAPTER I 

some assertive attitude can be adopted/ under which 
the relative determinates would be affirmed, denied, 
doubted, etc. We may indicate the nature of a given 
category by assigning the determinables involved in its 
construction. Using capital letters for determinables 
and corresponding small letters for their determinates 
(distinguished amongst themselves by dashes), the major 
premiss of the syllogism would assume the following 
form : Every MP is p if m ; where the determinables 
M and P serve to define the category so far as required 
for the syllogism in question. Here we substitute for 
the vague word thing previously employed, the symbol 
MP to indicate the category of reference ; namely, that 
comprising substantives of which some determinate 
character under the determinables Yfcf and P can be pre 
dicated. The statement that the given thing is MP is 
redundant where M and P are determinables to which 
the given thing belongs ; for the thing could not be given 
either immediately or in an act of construction except 
so far as it was given under the category defined by these 
determinables. Hence any genuine act of characterisa 
tion of the thing so given would consist in giving to 
these mere determinables a comparatively determinate 
value. For example, it being assumed that the given 
thing is MP, we may characterise it in such determi 
nate forms as m and /, m or/, p if m? not both/ 
and m where the predication of the relative determi 
nates m and p would presuppose that the object had 
been constructed under MP. In defining the function 
of a proposition to be to characterise relatively deter- 
minately what is given to be characterised, we now see 
that what is given is not given in a merely abstract 



INFERENCE IN GENERAL 17 

sense, but in being given the determinables which 
have to be determined are already presupposed. 

8. We may now show more clearly why the force 
of the term every is distinct from that of the term 
if; and how, in the syllogism, the two corresponding 
principles of inference are both involved. The major 
premiss having been formulated in terms of the deter 
minables M and P, the whole argument will assume 
the following form: 

(a) Every MP is/ if m, 

from which we infer, by the applicative principle alone: 

(b) The given MP is/ if m. 
Next we introduce the minor, viz. 

(c) The given MP is m, 

and finally infer, by the implicative principle alone: 

(d) The given MP is/. 

Now if we held that the inference from (a) to (b) re 
quired the implicative principle as well as the applica 
tive, so that a minor premiss The given thing is MP 
must be interpolated, the syllogism would assume the 
following more complicated form : 

(a) Everything is/ if m if MP (the reformulated 
major). 

. . (b) The given thing is / if m if MP (by the 
applicative principle alone). 
Next we introduce as minor 

(c) The given thing is MP. 

. . (d) The given thing is / if m (by the implicative 
principle alone); 

finally, introducing the original minor, viz. 

(e) The given thing is m. 

. . (/) The given thing is / (by the implicative prin 
ciple alone). 

J. L. II 2 



i8 CHAPTER I 

Now this lengthened analysis of the syllogism, while 
involving the implicative principle twice, involves as 
well as the applicative principle the introduction of a 
new minor, viz. that the given thing is MP, which hints 
at the doubt whether what is given is given as MP. 
But if this were a reasonable matter of doubt requiring 
explicit affirmation, on the same principle we might 
doubt whether what is given is a thing/ in some more 
generic sense of the word thing. If this doubt be ad 
mitted, the syllogism is resolved into three uses of the 
implicative principle, with two extra minor premisses. 
Such a resolution would in fact lead by an infinite regress 
to an infinite number of employments of the implicative 
principle. To avoid the infinite regress we must es 
tablish some principle for determining the point at 
which an additional minor is not required. The view 
then that I hold is not merely that what is given is a 
thing in the widest sense of the term thing, but that 
what is given is always given as demanding to be 
characterised in certain definite respects e.g. colour, 
size, weight; or cognition, feeling, conation and that 
therefore such a proposition as The given thing is 
MP is presupposed in its being given, i.e. in being 
given, it is given as requiring determination with respect 
to these definite determinables M and P. The above 
formulation, therefore, in which the syllogism is resolved 
into a process involving the applicative and the impli 
cative principles each only once, is logically justified; 
for it brings out the distinction between the function of 
the term every as leading to the employment of the 
applicative principle alone, and the function of if as 
leading to the employment of the implicative principle 



INFERENCE IN GENERAL 19 

alone; and furthermore it distinguishes between the 
process in inference which requires the applicative prin 
ciple alone from that which requires the implicative as 
well as the applicative principle. 

The distinction between the cases in which the im 
plicative principle can or cannot be dispensed wTEFi 

depends, so far, upon whether the subject-term of the f( 
universal stands for a logical category or not. But we 
may go further and say that, even if the subject of the 
universal is not a logical category, provided that it is 
definable by certain determinables, and that the subject 
of the conclusion is only apprehensible under those 



determinables, then again the use of the implicative 
principle may be dispensed with. For example: All 
material bodies attract; therefore, the earth attracts. 
Here the term material body is of the nature of a 
category in that it can only be defined under such de 
terminables as continuing to exist and occupying some 
region of space ; furthermore the earth is constructively 
given under these determinables: hence a proposed 
minor premiss to the effect that the earth is a material ^ ^ 
body is superfluous, and the above inference involves 
only the applicative principle. Again All volitional acts 
are causally determined; therefore, Socrates drinking (f>* f 
of hemlock was causally determined. Here the subject < 
of the conclusion is constructively given under the de 
terminables involved in the definition of volitional act, 
which again justifies the use of the applicative principle 
alone. As a third example : Every denumerable aggre 
gate is less than some other aggregate: therefore, an 
aggregate whose number is K is numerically less than 
some other aggregate. Here the construction of the 

22 fV* >. 



U 



20 CHAPTER I 

notion of a class whose number is K involves its being 
denumerable, so that the given inference again ^ re 
quires only the immediate employment of the applica 
tive principle. 

9. Incidentally the above analysis of the major 
premiss Every MP is p if m (or still more simply, 
Every M is m, which may sometimes be true; or 
again, of the minor premiss The given MP is m or 
The given MP is / ) accounts for the insistence by 
certain philosophers, notably Mr Bradley, that every 
proposition employs the relation of identity; i.e. that 
the adjective involved in the subject is the same as that 
involved in the predicate. This philosophical sugges 
tion is, I hold, true, in the sense that the adjectival 
deter minable in the subject is the same as that in the 
predicate; but the latter is a further determination of 
the former. Now, in this admission that the relation of 
identity of subject to predicate is involved in the general 
categorical proposition, I am not in any way with 
drawing what was maintained as regards identity in my 
analysis of the proposition. For the identity which I 
denied was (as it has been expressed) identity in deno 
tation with diversity of connotation, i.e. substantival 
identity with adjectival diversity. The identity I have 
accepted above is identity of an adjectival factor in the 
subject with an adjectival faster in the predicate. More 
over I should still deny that the proposition asserts this 
identity, and maintain that it simply presupposes it, in 
just the same way as a proposition presupposes the 
understanding of the meaning of the terms involved 
without asserting such meaning. 

10. We have discussed the case in which a minor 



INFERENCE IN GENERAL 21 

premiss may be dispensed with, namely that in which 
a certain mode of using the applicative principle is 
sufficient without the employment of the implicative. 
We will now turn to a complementary discussion of the 
case in which there is unnecessary employment of the 
applicative principle, entailed by the insertion of what 
may be called a redundant major premiss. It will be 
convenient to call the redundant minor premiss a sub- 
minor, and the redundant major premiss to which we 
shall now turn a super-major. In this connection I 
shall introduce the notion of a formal principle of in 
ference, which will apply, not only to inferences that are 
strictly formal, but also to inferences of an inductive 
nature, for which the principle has not at present been 
finally formulated and must therefore be here expressed 
without qualifying detail. The discussion will deal with 
cases in which the relation of premiss or premisses to 
conclusion is such that the inference exhibits a formal 
principle. 

We shall illustrate the point first by taking the 
principle of syllogism, and next, the ultimate (but as yet 
unformulated) principle of induction. As regards the 
syllogism, taking / and q to represent the premisses 
and r the conclusion, we may say that the syllogistic 
principle asserts that provided a certain relation holds 
between the three propositions p, q, and r, inference 
from the premisses p and q alone will formally justify 
the conclusion r. Now it might be supposed that this 
syllogistic principle constitutes in a sense an additional 
premiss which, when joined with p and q, will yield a 
more complete analysis of the syllogistic procedure. 
But on consideration it will be seen that there is a sort 



22 CHAPTER I 

of contradiction in taking this view : for the syllogistic 
principle asserts that the premisses p and q are alone 
sufficient for the formal validity of the inference, so that, 
if the principle is inserted as an additional premiss co 
ordinate with p and q, the principle itself is virtually 
contradicted. In illustration we will formulate the syllo 
gistic principle : 

What can be predicated of every member of a class, 
to which a given object is known to belong, can be pre 
dicated of that object/ 

Now, taking a specific syllogism : 

Every labiate is square-stalked, 
The dead-nettle is a labiate, 
.*. The dead-nettle is square-stalked, 

if we inserted the above-formulated principle as a pre 
miss, co-ordinate with the two given premisses, with a 
view to strengthening the validity of the conclusion, 
this would entail a contradiction; because the principle 
claims that the two premisses are alone sufficient to 
justify the conclusion The dead-nettle is square-stalked/ 
Now the same holds, mutatis mutandis, of any pro 
posed ultimate inductive principle. Here the premisses 
are counted not as two but as many, and summed up 
in the single proposition All examined instances charac 
terised by a certain adjective are characterised by 
a certain other adjective ; and the conclusion asserted 
(with a higher or lower degree of probability) predi 
cates of all what was predicated in the premiss of 
all examined. Now, in accordance with the inductive 
principle, the summary premiss is sufficient for asserting 
the unlimited universal (with a higher or lower degree 
of probability). To insert this principle, as an additional 



INFERENCE IN GENERAL 23 

premiss co-ordinate with the summary premiss, would, 
therefore, virtually involve a contradiction. In illustra 
tion, we will roughly formulate the inductive principle : 

What can be predicated of all examined members 
of a class can be predicated, with a higher or lower 
degree of probability, of all members of the class. 

Now, taking a specific inductive inference: 

All examined swans are white. . . With a higher 
or lower degree of probability, all swans are white/ 

if we inserted the above-formulated inductive principle 
as a premiss, co-ordinate with the summary premiss All 
examined swans are white, with a view to strengthening 
the validity of the conclusion, this would entail a con 
tradiction ; because the principle claims thatthis summary 
premiss is alone sufficient to justify the conclusion that 
With a higher or lower degree of probability, all swans 
are white/ 

We may shortly express the distinction between a 
principle and a premiss by saying that we draw the 
conclusion from the premisses in accordance with (or 
through) the principle. In other words, we immediately 
see that the relation amongst the premisses and con 
clusion is a specific case of the relation expressed in the 
principle, and hence the function of the principle is to 
stand as a universal to the specific inference as an in 
stance of that universal : where the latter may be said 
to be inferred from the former (if there is any genuine 
inference) in accordance with the Supreme Applicative 
principle. For example : from x =y and y = z, we may 
infer x = z. This form of inference is expressed, in 
general terms, in the Principle : Things that are equal 
to the same thing are equal to one another. Now, here, 



a.. 



24 CHAPTER I 

the two premisses x--=y andy = z are alone sufficient 
for the conclusion x = z\ the conclusion being drawn 
from the two premisses through or in accordance with 
the principle which states that the two premisses are 
alone sufficient to secure validity for the conclusion. 
The principle cannot therefore be added co-ordinately 
to the premisses without contradiction. Moreover the 
above-formulated principle (which expresses the tran 
sitive property of the relation of equality) cannot be 
subsumed under the syllogistic principle. In the same 
way the syllogistic or inductive principle may be called 
a redundant or super-major, because it introduces a mis 
leading or dispensable employment of the applicative 
principle. 

ii. There is a special purpose in taking the in 
ductive and syllogistic principles in illustration of super- 
majors, for many logicians have maintained that any 
specific inductive inference does not rest on an inde 
pendent principle, but upon the syllogistic principle 
itself; in other words, they have taken syllogism to 
exhibit the sole form of valid inference, to which any 
other inferential processes are subordinate. Now it is 
true that the inductive principle could be put at the 
head of any specific inductive inference, and thus be 
related to the specific conclusion as the major premiss 
of a syllogism is related to its conclusion ; but the same 
could be said of the syllogistic principle : namely that it 
could be put at the head of any specific syllogistic in 
ference to which it is related in the same way as the 
major premiss of a syllogism is related to its conclusion. 
But, if we are further to justify the specific inductive 
inference by introducing the inductive principle, then, 



INFERENCE IN GENERAL 25 

by parity of reasoning, we should have to introduce the 
syllogistic principle further to justify the specific syllo 
gistic inference. But in the case of the syllogism this 
would lead to an infinite regress as the following illus 
tration will show. Thus, taking again as a specific 
syllogism, that 

from (p) All labiates are square-stalked 

and (q] The dead-nettle is a labiate 
we may infer (r) The dead-nettle is square-stalked, 
and, adding to this as super-major the syllogistic 
principle, namely (a), we have the following argument : 

(a) For every case of M, of 5 and of P: the inference 
every M is P, and S is M, .-. S is P is valid. 

(6) The above specific syllogism is a case of (a). 

(c) .*. The specific syllogism is valid. 
But here, in inferring from (a) and (b) together to (c), 
we are employing the syllogistic principle, which must 
stand therefore as a super- major to the inference from 
(a) and (b) together to (c), and therefore as super-super- 
major to the specific inference from p and q to r. This 
would obviously lead to an infinite regress. 

We may show that a similar infinite regress would 
be involved if we introduced, as super-major, the in 
ductive principle, by the following illustration. Taking 
again as a specific inductive inference that from All 
examined swans are white we may infer with a higher 
or lower degree of probability that All swans are 
white ; and adding to this as super-major the in 
ductive principle, namely (a), we have the following 
argument : 

(a) For every case of M and of P: from every 
examined M is P, we may infer, with a higher or lower 
degree of probability, that every M is P \ 



;v 



26 CHAPTER I 

(b] The above specific induction is a case of (a). 

(c) . . The specific induction is valid. 

But, here we may argue in regard to this (a), (b), (c) as 
in the case of the previous (a), (6), (c). Thus, by in 
troducing the inductive principle as a redundant major 
premiss, we shall be led as before, by an infinite regress, 
to a repeated employment of the syllogistic principle. 

This whole discussion forces us to regard the in 
ductive and syllogistic principles as independent of one 
another, the former not being capable of subordination 
to the latter; for we cannot in any way deduce the in 
ductive principle from the syllogistic principle. Those 
who have regarded the syllogistic principle as ultimately 
supreme, have in fact arrived at this conclusion by noting 
that, as shown above, the inductive principle could be 
introduced as a major for any specific inductive inference, 
in which case the inference would assume the syllogistic 
form (a), (d), (c). But this in no way affects the supremacy 
of the inductive principle as independent of the syllo 
gistic. 



CHAPTER II 

THE RELATIONS OF SUB-ORDINATION AND CO-ORDI 
NATION AMONGST PROPOSITIONS OF DIFFERENT 
TYPES 

i. IN the previous chapter we have shown that the 
syllogism which establishes material conclusions from 
material premisses involves the alternate use of the 
Applicative and Implicative principles. Now these two 
principles, which control the procedure of deduction in 
its widest application, are required not only for material 
inferences, but also for the process of establishing the 
formulae that constitute the body of logically certified 
theorems. All these formulae are derived from certain 
intuitively evident axioms which may be explicitly 
enumerated. It will be found that the procedure of 
deducing further formulae from these axioms requires 
only the use of the Applicative and Implicative prin 
ciples; these, therefore, cover a wider range than that 
of mere syllogism. But a final question remains, as to 
how the formal axioms are themselves established in 
their universal form. By most formal logicians it is 
assumed that these axioms are presented immediately 
as self-evident in their absolutely universal form ; but 
such a process of intuition as is thereby assumed is 
really the result of a certain development of the reasoning 
powers. Prior to such development, I hold that there 
is a species of induction involved in grasping axioms in 
their absolute generality and in conceiving of form as 



28 CHAPTER II 

constant in the infinite multiplicity of its possible appli 
cations. We therefore conclude that behind the axioms 
there are involved certain supreme principles which bear 
to the Applicative and Implicative principles the same 
relation as induction in general bears to deduction ; and, 
even more precisely, that these two new principles may 
be regarded as inverse to the Applicative and Impli 
cative principles respectively. This being so, it will be 
convenient to denominate them respectively, Counter- 
applicative andCounter-implicative. It should be pointed 
out that whereas the Applicative and Implicative prin 
ciples hold for material as well as formal inferential 
procedure, the Counter-principles are used for the 
establishment of the primitive axioms themselves upon 
which the formal system is based. We will then pro 
ceed to formulate the Counter-principles, each in imme 
diate connection with its corresponding direct principle. 
2. The Applicative principle is that which justifies 
the procedure of passing from the asserting of a pre 
dication about every to the asserting of the same 
predication about any given. Corresponding to this, 
the Counter-applicative principle may be formulated: 

1 When we are justified in passing from the asserting 
of a predication about some one given to the asserting 
of the same predication about some other, then we are 
also justified in asserting the same predication about 
every! 

Roughly the Applicative principle justifies inference 
from every to any, and the Counter-applicative 
justifies inference from any to every ; but whereas 
the former principle can be applied universally, the 
latter holds only in certain narrowly limited cases; and, 



SUB-ORDINATION AMONGST PROPOSITIONS 29 

in particular, for the establishment of the primitive 
formulae of Logic. These cases may be described as 
those in which we see the universal in the particular, 
and this kind of inference will be called intuitive in 
duction, because it is that species of generalisation in 
which we intuite the truth of a universal proposition in 
the very act of intuiting the truth of a single instance 1 . 
Since intuitive induction is of course not possible in 
every case of generalisation, we have implied in our 
formulation of the principle that the passing from any 
to every is justified only when the passing from any 
one to any other is justified. Now there are forms of 
inference in which we can pass immediately from any 
one given case to any other ; if it were not so, the 
principle would be empty. For instance, we may illus 
trate the Applicative principle by taking the formula: 
For every value of/ and of q, "p and q" would imply 
"/", from which we should infer that thunder and 
lightning would imply thunder. If now we enquire 
how we are justified in asserting that for every value 
of / and of q, p and q would imply /, the answer 
will supply an illustration of the Counter-applicative 
principle. Thus, in asserting that "thunder and light 
ning" would imply "thunder" we see that we could 
proceed to assert that "blue and hard" would imply 
"blue", and in the same act, that "/ and q" would 
imply "/" for all values of/ and of q? 

3. The second inverse principle to be considered is 
the Counter-implicative. Before discussing this inverse 
principle, it will be necessary to examine closely the 

1 This is a special case of intuitive induction, the more general 
uses of which will be examined in Chapter VIII. 



30 CHAPTER II 

Implicative principle itself, which may be provisionally 
formulated: Given that a certain proposition would 
formally imply a certain other proposition, we can validly 
proceed to infer the latter from the former. Now we 
find that the one positive element in the notion of 
formal implication is its equivalence to potentially valid 
inference, and that there is no single relation properly 

called the relation of implication. We must therefore 

f 

bring out the precise significance of the Implicative 
/H principle by the following reformulation: There are 
certain specifiable relations such that, when one or 
rw*&U *k& other of these subsists between two propositions, we 
may validly infer the one from the other. From the 
enunciation of this principle we can pass immediately 
to the enunciation of its inverse the Counter-implica- 
tive principle : 



When we have inferred, with a consciousness of 
validity, some proposition from some given premiss or 
premisses, then we are in a position to realise the specific 
form of relation that subsists between premiss and con 
clusion upon which the felt validity of the inference 
depends. 

Here, as in the case of the Counter-applicative principle, 
we must point out that there are cases in which we in 
tuitively recognise the validity of inferring some con 
crete conclusion from a concrete premiss, before having 
recognised the special type of relation of premiss to 
conclusion which renders the specific inference valid; 
otherwise the Counter-implicative principle would be 
empty. In illustration, we will trace back some accepted 
relation of premiss to conclusion, upon which the validity 
of inferring the one from the other depends; and this 



SUB-ORDINATION AMONGST PROPOSITIONS 31 

will entail reference to a preliminary procedure in ac 
cordance with the Counter-applicative principle; for 
every logical formula is implicitly universal. Thus we 
might infer, with a sense of validity from the information 
Some Mongols are Europeans and from this datum 
alone, the conclusion Some Europeans are Mongols. 
We proceed next in accordance with the Counter-appli 
cative principle to the generalisation that the inference 
from Some M is P to * Some P is M is always valid. 
Finally we are led, in accordance with the Counter- 
implicative principle, to the conclusion that it is the re 
lation of converse particular affirmatives that renders 
the inference from Some M is P to Some P is M 
valid. 

4. We have regarded the intuition underlying the 
Counter-applicative principle as an instance of seeing 
the universal in the particular ; and correspondingly the 
intuition underlying the Counter-implicative principle 
may be regarded as an instance of abstracting a common 
form in diverse matter. But the direct types of intuition 
operate over a much wider field than the Counter-appli 
cative and Counter-implicative principles : for, whereas 
the twin inverse principles operate only in the estab 
lishment of axioms, the direct types of intuition 
are involved wherever there is either universality or 
form. These direct types of intuition have been ex 
plicitly recognised by philosophers ; but the still more 
purely intuitive nature of the procedure conducted in 
accordance with the twin inverse principles accounts for 
the fact that these principles have hitherto not been 
formulated by logicians. Moreover the point of view 
from which the inverse principles have been described 



32 CHAPTER II 

and analysed is purely epistemic \ and the epistemic 
aspect of logical problems has generally been ignored 
or explicitly rejected by logicians. It follows also from 
their epistemic character that these principles, unlike 
the Applicative and Implicative principles of inference, 
cannot be formulated with the precision required for a 
purely mechanical or blind application. 

5. The operation of these four supreme principles 
is best exhibited by means of a scheme which comprises 
propositions of every type in their relations of super-, 
sub-, or co-ordination to one another. We propose, 
therefore, to devote the remainder of this chapter to 
the construction and elucidation of such a scheme. 

I. Super or dinate Principles of Inference. 

\a. The Counter-applicative and Counter-impli- 
cative. 

\b. The Applicative and Implicative. 

II. Formulae-, i.e. formally certified propositions 
expressible in terms of variables having general 
application. 

\\a. Primitive formulae (or axioms) derived 
directly from II I a in accordance with la. 

I 1 b. Formulae successively derived from 1 1 a by 

means of \b. 

III. Formally Certified Propositions expressed in 
terms having fixed application. 

III a. Those from which II a are derived by use 

of the principles I a. 

lllb. Those which are derived from lib by use 
of the Applicative principle I b. 

IV. Experientially Certified Propositions. 
IV a. Data directly certified in experience. 



SUB-ORDINATION AMONGST PROPOSITIONS 33 

IV 6. Concrete conclusions inferred from IV a by 
means of implications of the type III, and 
therefore established in accordance with the 
Implicative principle, 16. 

I. The highest type consists of those principles 
under one or other of which every inference is sub 
ordinated. These superordinate principles consist of 
I a : the Counter-applicative and Counter-implicative, 
to which intuitional inferences are subordinated; and 
of \b\ the Applicative and Implicative, to which de 
monstrative inferences are subordinated. \a are those 
principles in accordance with which the primitive for 
mulae (or axioms) of Logic are established. But the 
choice of logical formulae that are accounted primitive 
is (within limits) arbitrary, and since any comparatively 
self-evident logical formula, instead of being exhibited 
as derivative, could be regarded as established directly 
in accordance with these inverse principles, their scope 
must not be restricted to the establishment of the more 
or less arbitrarily selected axioms. It will be found later, 
when we discuss the types of proposition in level III, 
that the content or material upon which the inverse prin 
ciples I a operate, is supplied by the propositions of type 
1 1 1 a. On the other hand, the Applicative and Implica 
tive principles \b stand in the relation of immediate 
superordination to the processes of inference by which 
from 1 1 a are derived 1 1 b, viz. the general formulae of 
deduction, induction, demonstration, probability, etc. 

1 1. The characteristic common to all the propositions 
on the second level is that they are formally certified, 
and are expressible in terms of variable symbols. They 
are theoretically infinite in number, and may be divided 

j. L. ii 3 



34 CHAPTER II 

into two groups, primitive and derivative ; but, as 
pointed out above, the line of demarcation between the 
two cannot be sharply drawn. Thus II a comprises a 
small number of primitive formulae which are directly 
established in accordance with the twin inverse principles 
la: for example, the commutative and associative laws, 
the laws of identity and of negation, the modus ponendo 
tollens, etc., or such of these as have been selected as 
primitive. Next, lib comprises an indefinite number 
of formulae successively derived from the primitive for 
mulae II#: for example, the dictum of the syllogism, 
and other more complicated logical formulae, as well as 
the rules of arithmetic and algebra. All the formulae of 
level 1 1 are implicitly universal in form ; and most of 
those that are logical (as distinct from mathematical) 
assert relations of implication. Each formula in II is 
derived from previously certified formulae, and ultimately 
from those in I la, the process of derivation being marked 
at each step by the relation therefore. Now wherever 
a previously certified relation of implication is used for 
deriving a new formula (in which case its implicans 
must also have been previously certified in order that 
its implicate may be derivatively certified) the procedure 
is conducted in accordance with the implicative principle, 
to which therefore all such cases of inference are to be 
subordinated. Again, the process of successive deriva 
tion of the formulae of 116 entails explicit recognition 
of the implicit universality of the formulae from which 
they are derived ; and this allows us, by means of the 
Applicative principle, to replace the illustrative symbols 
occurring in an earlier formula by any other symbols, 
in order to derive a new formula. 



SUB-ORDINATION AMONGST PROPOSITIONS 35 

III. The third level contains formally certified pro 
positions expressed entirely in linguistic terms of fixed 
application ; and, like its predecessors, is to be divided 
into two sections, the division being made on precisely 
the same grounds as that between 110 and 1 1 & Thus 
the propositions of Ilia constitute the intuited material 
for deriving 1 1 a in accordance with the inverse prin 
ciples la; and the propositions of III^ are exhibited 
as derived from 116 in accordance with the applicative 
principle I b l . It will be seen, however, that the relation 
of Illtfto Hid differs from that of llato Il^in that the 
two parts of I II are not inferentially connected, as are 
those of II. The propositions comprised in II I b are 
obtained from 116 by substituting words with fixed 
application for the variable symbols ; these proposi 
tions, then, are specialised instances of the general 
formulae which constitute the second level, and are 
established from them in accordance with the appli 
cative principle alone. Any logical text-book teems 
with examples of this procedure, where instances under 
such formulae as the modus tollendo tollens, or the 
syllogistic dictum are represented in words with fixed 
application, and then exhibited as derived (in ac 
cordance with the applicative principle) from the appro 
priate general formula. It is usual in these cases, how 
ever, to exhibit the conclusion as being inferred from 
the premisses, thus leading the reader to suppose that 
it is the conclusion which has been formally certified, 
whereas, properly speaking, what has been formally 
certified is the relation of implication of premisses to 

1 Hence the point of division between Ilia and Hlb cannot be 
precisely indicated. 

32 



3 6 CHAPTER II 

conclusion. It will be found below that this distinction 
between implication and inference is the essential con- 
i sideration in comparing \lld with IV b. 

IV. The fourth and lowest level consists of experi- 
entially certified propositions expressed in concrete 
terms ; and again this level must be divided into two 
sections, viz. IV a the primitives and IV b the deriva 
tives, these two sections standing in a relation to one 
another which in every respect agrees with the rela 
tion of II a to II & Thus the propositions comprised 
in 1Mb are successively derived from experiential 
propositions that have been previously certified, and 
ultimately derived from the primitive experiential data 
which constitutes IV a. And again, as in the case of 
formally certified propositions, here, in that of ex- 
perientially certified propositions, the point of division 
between the primitives and derivatives is not precisely 
fixed ; the primitives of IV, like those of II, are sup 
posed to be intuitively accepted, i.e. in this case per 
ceptually guaranteed; but philosophers do not agree 
on the question of the kind and range of experiences 
that can be regarded as in this case immediate. More 
over, as regards experiential propositions admittedly 
derivative and not primitive, no logician or philosopher 
has as yet been able to show how they can be exhibited 
as derived ultimately from absolutely primitive data of 
experience. Hence, in expounding the logical nature 
of the propositions in this lowest level, attention must 
be chiefly directed to the mode in which any admittedly 
derived proposition is inferred from some previously 
certified proposition, without enquiring too closely as 
to the mode in which the previous certification had 



SUB-ORDINATION AMONGST PROPOSITIONS 37 

been conducted, or whether this certification could 
properly be called perceptually immediate. The mode 
of deriving an experiential conclusion from experi- 
entially certified premisses may be explained quite 
briefly ; the former is derived directly from the latter 
by means of some implication of type III, of which the 
implicans is composed of the previously certified pre 
misses and the implicate is the conclusion required. 
Since in this process a relation of implication is trans 
formed into the relation therefore, it is obvious that 
the implicative principle alone is employed. But, to 
complete the exposition, we must trace the process of 
derivation one stage further back, namely to the general 
formulae of line II. Thus, while any conclusion in IV$ 
is directly derived from premisses I V a by means of an 
implicative proposition of the type III, and so far 
employs the implicative principle alone ; yet, since any 
proposition of type III is itself derived from some 
formula of type 1 1 in accordance with the applicative 
principle alone, it follows that both these principles are 
jointly involved in deriving experiential conclusions 
from experiential data. This mode of derivation is 
illustrated in any text-book example of a concrete 
syllogism, where from previous experiential certification 
of the premisses we infer the experiental certification 
of the conclusion. For the sake of variety we will 
choose, for illustrating the processes of deriving any 
conclusion IV<, the formula of pure induction, which, 
as was maintained in the preceding chapter, must be 
included amongst the formulae constituting the second 
level. Take for instance as premiss : Every examined 
case of an acquired characteristic is non-transmitted. 



38 CHAPTER II 

This datum is regarded not, of course, as a mere sum 
mary of directly given experiences, but as the product 
of various constructive and inferential processes which 
may be supposed ultimately to be based on sense-data. 
Now by means of the concrete implication that * every 
examined case of an acquired characteristic being non- 
transmitted would imply, with a certain degree of pro 
bability, that no acquired characteristic is transmissible/ 
conjoined with the certified fact that * in all examined 
cases acquired characteristics are non-transmitted, we 
infer the conclusion that with a certain degree of 
probability no acquired characteristic is transmissible/ 

In this fourth line, we are representing propositions 
as proved, or as validly asserted on the basis of ex 
periential knowledge, and this suggests an ambiguity 
in the use of the term ground which is sometimes 
applied in philosophy to the experiential data which 
may be said to be co-ordinate with the experiential con 
clusions; the same term ground being also applied to 
the logical formulae of induction or deduction which 
are superordinate to the experiential data and con 
clusions. This ambiguity in the use of the term is 
removed by thus recognising the distinction between 
superordinate and co-ordinate. 

6. In further elucidation of the scheme, we will 
show what exactly is involved in level II, where em 
phasis has been put upon the variable symbols. In 
logical text-books we find that an inference or implica 
tion is expressed in terms of variable symbols, such as 
S, M y and P, and this is always supplemented by a 
formula expressed entirely linguistically, but which is 
its mere equivalent. For example, it may be first 



SUB-ORDINATION AMONGST PROPOSITIONS 39 

asserted that Every P is Q would imply that some Q 
is P y ; and here the assertion of implication is under 
stood as being implicitly universal, i.e. that it holds for 
all values of P and Q. This is usually supplemented by 
the so-called Rule for the Conversion of A viz. that 
Any universal affirmative proposition would imply the 
particular affirmative obtained by interchanging subject 
and predicate terms. But this is merely an alternative 
formulation, and is not related to the former as a 
universal to its instance. We see therefore that the 
formulae of level II are not necessarily expressed in 
terms of variables, but may be expressed with precise 
equivalence in linguistic terms only. The possibility of 
this linguistic formulation depends upon the invention 
of a technical terminology which employs such terms as 
subject, predicate, conversion, universal, proposition, 
etc. The reason why what is called symbolic logic 
requires the employment of variable symbols is essen 
tially because the logical formulae which it establishes 
are so complicated that a terminology could hardly be 
invented for dealing with them. There is therefore no 
difference of principle involved in the employment of 
variable symbols by symbolic logic and the employ 
ment of technical linguistic terms by ordinary logic. 
By the employment of the technical terminology of 
logic the variables entering into any formula are elimi 
nated en bloc, leaving the formula with the same range 
of universality as before. In contrast with this, a pro 
position of level III, being obtained from level II by 
replacing each of the several variables by a particular 
word of fixed application, constitutes a single instance 
of the general formula. For instance, that every 




40 CHAPTER II 

trespasser will be prosecuted would imply that some 
prosecuted person is a trespasser 1 / is a specific assertion 
obtained by the applicative principle from the universal 
formula of conversion adduced above. 

This last discussion of the distinction and connection 
between the use of variable symbols and that of linguistic 
terminology, points to certain respects in which the 
methods of symbolic logic differ, and others in which 
they agree with those of ordinary logic a topic which 
will be treated at greater length in the following chapter. 

1 This illustration is chosen in order incidentally to suggest that 
the text-books are not always infallible, the form of implication in 
question being at least dubious. 



CHAPTER III , 

SYMBOLISM AND FUNCTIONS 

V 

i. THE value of symbolism, as is universally re 
cognised, is due to the extreme precision which its 
employment affords to the process of logical demonstra 
tion. As a language it differs from all ordinary languages 
in three respects, viz. systematisation, brevity and 
exactness; and in these respects differs from all other 
languages in a way in which they do not differ from one 
another. 

Now, when we examine the language of symbolism, 
we find that symbols are of two fundamentally distinct 
kinds, which I propose to call illustrative and shorthand. 
In such familiar logical forms as S is /Y Every M is 
/Y etc., S, M, P, exemplify illustrative symbols. Thus 
an illustrative symbol is represented by a single letter 
chosen from some alphabet. Shorthand symbols, on 
the other hand, are mere substitutes for words, and 
serve the obvious purpose of saving time in reading 
and space in writing. Some of them, in fact, are literal 
abbreviations, such as rel. for relative/ prop. for 
1 proposition/ indiv. for individual/ Others again are 
arbitrarily shaped marks standing for simple words such 
as not, and, or, if, is, identical with. A third kind of 
shorthand symbol is one introduced in the course of a 
symbolic calculus, and defined in terms of combinations 
of other shorthand symbols, and ultimately in terms 
of the simple symbols introduced at the outset. So far, 



42 CHAPTER III 

a shorthand symbol has all the characteristics of a word 
or a word-complex only differing from these in satisfying 
the essential symbolic requirements of systematisation, 
brevity and exactness. In one respect, however, these 
symbols differ from such word-complexes as that man/ 
the river, Mr Smith, this experience, my present 
purpose, in that these latter have a meaning or appli 
cation not universally fixed but determined only by 
means of context ; whereas the symbols of Logic have 
an unalterable meaning wholly independent of context, 
and resemble rather, such word-complexes as rational 
animal, loud, hard, church, differing from these 
however in being strictly unambiguous. Ordinary Logic 
generally dispenses with symbols of this kind the most 
familiar exception being Dr Keynes s SaP, which is 
shorthand for Every 5 is P, etc. On the other hand, 
Dr Keynes himself shows, in his Appendix C, how 
certain complicated problems, previously relegated to 
Symbolic Logic, can be solved without recourse to 
shorthand symbols, illustrative symbols only being in 
troduced. 

Now an important character of the shorthand symbol 
is that its constancy is logical or formal and not expe 
riential or material. A formal constant is one whose 
meaning is to be understood by the logician as such; 

that is to say, logic pronounces it either as indefinable 

because understood without requiring definition or as 
definable in terms of logically understood constants 
alone. The following is a rough classification of formal 
constants expressed in ordinary language : ( i ) the articles 
or applicatives; a, the, some, etc. (2) the negative not\ 
and the conjunctions and, or, if, etc. (3) the copula is\ 



SYMBOLISM AND FUNCTIONS 43 

and certain prepositions such as of, to, in some of their 
meanings. (4) certain relations such as identical with, 
comprised in. (5) such modal adjectives as true, false, 
probable, etc. Formal constants are to be contrasted 
with material in that the meanings of the latter are to 
be understood in terms of ideas or conceptions outside 
the sphere of logic. The division between formal and 
material constants, i.e. between what is and what is not 
required for the understanding of logical principles, can 
ultimately be rendered precise only after a complete 
logical system has been constructed. For instance, 
numerical adjectives such as two andy^w would have 
been pronounced as merely material at the stage at 
which the logical system had not been carried on into 
its mathematical developments. Ideas that are imme 
diately recognised as material relatively to the essentials 
of logic are those of sense-qualities, or of the properties 
and characteristics of physical and mental entities. 
Temporal and spatial relations, being in one aspect sub- 
sumable under the conceptions of order, would, so far, 
be called formal or logical, but, inasmuch as these rela 
tions actually have a specific over and above their 
generic significance, they must be treated also as 
having an experiential or material source. The same 
holds of the determinates of a determinable, inasmuch 
as experience is required in order to present to the 
mind any single determinable and to distinguish one 
determinable from another, whilst the discussion of the 
formal relations of incompatibility, order, etc., between 
determinates under any determinable is purely logical. 
Since shorthand symbols and the words or word- 
complexes of ordinary language function in the same 



44 CHAPTER III 

way, there is no essential difference between them if 
we take the symbols or words in isolation apart from 
consideration of the mental processes involved in their 
use. The psychological distinction is not between 
words and symbols as such but between the linguistic 
and the symbolic mode in which we think with their 
assistance. Thus, in linguistic thought, the words or 
symbols presented in imagination or vocalisation are the 
means or instruments by which we can attend to or 
think about the objects for which they stand. On the 
other hand, such a phrase as Waterloo was fought in 
1815 might illustrate the symbolic use of language 
which consists not in thinking about the objects for 
which the words stand but in mentally rehearsing the 
language in which propositions previously accepted have 
been expressed. Now the previous acceptance of these 
propositions must have entailed genuine processes of 
thinking; but, when they are recalled, we need not 
repeat these mental processes. It is in this way that 
the symbolic is distinguished from the linguistic use of 
words or symbols. In the latter, we are thinking by the 
use of words; whereas, in the former, recall of the 
words serves merely as a substitute for a previous act 
of thought 1 . 

2. These preliminary considerations bring us to 
the question : What actually happens in the mind of the 
symbolist, when he is either constructing or intelligently 
following the formulae of a symbolic calculus? In the 
first place, the axioms of the calculus can only be es 
tablished by the use of what I have called the Counter- 

1 This subject will be found to be more fully treated in Dr Stout s 
Analytic Psychology. 



SYMBOLISM AND FUNCTIONS 45 

applicative and Counter-implicative principles, and here 
genuine thought is required on the part of the symbolist. 
In the second place, the construction of any symbolic 
calculus involves the procedure of inference; and this 
is conducted always in accordance with the Applicative 
principle, and, in the case of the logical calculus, also in 
accordance with the Implicative principle. When pro 
ceeding in accordance with these principles, the sym 
bolist is actually thinking; he is not merely recalling 
verbal formulae in which the results of previous acts of 
thought have been expressed. In thp tfy rfl place, pven 
a perfectly constructed symbolic system would need to 
introduce some axioms, as also some propositions derived 
from axioms, that can only be expressed in non-symbolic 
terms. This necessary recourse to ordinary language in 
developing a deductive system shows that direct atten 
tion to meanings, presented linguistically, is entailed in 
the intelligent following of even a professedly symbolic 
exposition. Lastly, the extent to which thought can be 
dispensed with, when working a calculus, depends very 
largely and essentially upon the extent to which the 
system requires what maybe called interpretation clauses 
such as when P stands for any proposition, or where 
x is to be understood as a variable and a as a constant. 
If the symbolic language is so constructed that a mini 
mum of interpretation clauses is required, then there is 
a corresponding minimum in the extent to which actual 
thinking is involved. But, however few interpretation 
clauses are required, the intelligent use of symbolic 
formulae cannot be reduced to a merely mechanical 
process. This will be still more apparent from an exami 
nation of the nature of a symbolic system in which both 



4 6 CHAPTER III 

shorthand and illustrative symbols enter in combination 
with one another. 

3. For this purpose we will further consider the 
characteristics of illustrative symbols. These, being 
nothing but arbitrarily chosen letters of the alphabet, 
differ from words of ordinary language in that they 
cannot be interpreted as standing for this rather than 
for that specific object or idea; and hence, in the nature 
of the case, have a variable application. The writer or 
reader of a symbolic system must always bear in mind, 
however, that the variability in application of an illus 
trative symbol in any given case is not wholly unre 
stricted, but is limited within an understood range. 
Thus a single letter used illustratively must be under 
stood to be restricted in one case (say) to any substan 
tive; in another (say) to any adjective; and in another 
again (say) to any proposition, these being the three 
most prominent categories to which illustrative symbols 
are applied. Symbolic devices may, indeed, be invented 
by which to distinguish one kind of symbol as appli 
cable to a substantive, and another kind to some other 
specific category; but the range of application to be 
understood by letters taken in combination could not be 
indicated by any such device. When single letters are 
bound together into a complex by means of logical 
constants, then a further act of intelligence is required 
in interpreting such complex. For example, under 
standing in the first place the letters p, q, r, to stand 
for propositions, such constructs as / and qj / or qj 
p if q? must \*z further interpreted as also constituting 
propositions. Thus, when a formula about any or all 
propositions has been established, we may proceed to 



SYMBOLISM AND FUNCTIONS 47 

apply it to any complex such as / and q or p if q* and 
so on in accordance with the Applicative principle, in 
asmuch as each of such complexes constitutes a pro 
position. Similarly, when such letters as x, s, t are 
understood to stand for substantives, and such letters 
as /, q, r to stand for adjectives, then a further act of 
intelligence is required to interpret such a complex as 
*s is/ as standing for a proposition. This presupposes 
that the logical analysis of the simple proposition into 
the form s is// where s is understood to stand for a 
substantive and p for an adjective, has been discussed 
and established in a preliminary account in which words 
and not symbols were employed. Prepositional signifi 
cance having been attached to this form of construct, a 
distinct act of intelligence is required when, in uniting 
say s is/ with t is q in some form of combination, 
the resulting construct is understood to stand for a pro 
position. As another example illustrating the need for 
intelligent activity in symbolic work, we may take the 
two prepositional forms s e p and *x i y! where e is 
shorthand for the copula is and t i t for is identical 
with. Not only must these forms be interpreted as 
standing for propositions, but the relation for which e 
stands must be understood to be different from that for 
which *i stands. In consequence, when these two forms 
occur, reference must be made to one set of established 
formulae for the one case, and to a different set for the 
other. The necessity for using this modicum of intelli 
gence is to be contrasted with the purely blind or 
mechanical process required of the reader or writer in 
making use of the formulae to which he refers; for, in 
this latter process, he need attach no significance to e 



4 8 CHAPTER III 

or to /, as each standing for its own specific relation. 
The examples adduced have been selected on the ground 
of their simplicity, but complex examples would have 
brought out more forcibly the importance of the distinc 
tion between the intelligent and the merely mechanical 
operations required in working a symbolic system. 

4. Now the variability that characterises illustra 
tive symbols constitutes a special feature of symbolism, 
and its further discussion requires the introduction of 
the notion function. This term is used by logicians 
and mathematicians in a sense quite unconnected with 
the biological meaning of the term. The notion of a 
function is closely connected with the notion of a con 
struct, but the former must be understood relationally, 
whereas this is not obviously the case with the term 
construct. Thus, we should speak of a certain construct 
as being a function of certain enumerated constituents. 
The notation for a function in general is/(#, b, c, ...) 
where a, b, c, ... stand for the constituents ; and where 
the order in which these constituents are written is 
essential, so that/(#, b, c, ...) is not necessarily equiva 
lent to/(#, c, b, ...). Thus any function of a, b, c, ... is a 
construct involving a,b,c,.... But, if this were all that 
could be said about a function, the term would have no 
special value, since it would be a mere synonym for 
construct involving. The importance of the notion 
of function lies in the fact that we may speak of the 
same function in reference to different constituents, 
whereas the same construct would of course entail the 
same constituents. Thus, if C be a certain construct 
involving a, b, c, ..., and if D be another construct 
involving p,q,r,..., then C is said to be the same 



SYMBOLISM AND FUNCTIONS 49 

function of a, 6, c, ... as is D of p, q, r, . . . , when the 
substitution of/ for a, q for b, r for c, etc., would render 
D identical with C. Thus, in order to decide as regards 
two constructs, whether they express the same or a 
different function, we must specify the constituents of 
which the construct is regarded as a function ; and, to 
avoid all possible ambiguity, all the constituents for 
which substitutions have to be made must be enu 
merated. To explain this necessity, it must be pointed 
out that a construct may involve, implicitly or explicitly, 
other constituents in addition to those of which it is to 
be regarded as a function. In order to indicate the 
sameness of function exhibited by different constructs, 
it is therefore essential to enumerate those constituents 
for which substitutions are contemplated. These con 
stituents will be called variants^, because it is these and 
these alone that have to be varied in order to obtain 
the different constructs that exhibit the same function. 
On the other hand, in exhibiting identity of function, 
terms entering into the construct that are not to be 
replaced by some other terms will be called constants or 
non- variants. Hence the distinction between a variant 
and a non-variant constituent of a construct has rele 
vance only to functional identity. Since a function and 
its variants are to be understood relationally to one 
another, we may speak of the variants for a certain 
function just as we speak of a function of certain 

1 The word variant is here and throughout used in place of the 
mathematically technical word argument^ partly in order to prevent 
confusion with the ordinary logical use of the latter word, and partly 
in order to bring out the distinction and connection between the 
notion of variant and that of variable. 

j. L. ii 4 



50 CHAPTER III 

variants. In a complicated symbolic system it is found to 
be convenient to use, in place of a singular or proper 
name, an illustrative symbol which, qua symbol, must 
be what is called variable. Variability is therefore the 
mark of an illustrative symbol as such, whereas the con 
trast between variant and non-variant holds not of a 
symbol but of that for which the symbol may stand ; 
and, as has been said, this latter contrast has no sig 
nificance apart from the notion of a function. 

5. In considering the constituents of a construct 
with a view of indicating which are to be variants and 
which non-variants for a function, we must first note 
the distinction between material and formal constituents. 
Now as regards the strictly formal constituents of a 
construct, logic never contemplates making substitu 
tions for these ; hence, in all applications of the notion 
of a function in reference to its variants, two cases 
only have to be considered ; (i) the function for which 
all the material constituents are treated as variants, 
and (2) the function for which some of the material 
constituents are treated as constants and others as 
variants in both cases the formal constituents being 
understood to be constants. When (i) all the material 
constituents are to be varied, then the function may be 
said to be formal ; and the form of a construct is a 
brief synonym for the formal function which it exhibits. 
But, when (2) some of the material constituents are to 
be constant, then the function will be said to be non- 
formal. It follows that, when two constructs can be 
said to exhibit the same formal function, their reduction 
to identity is effected by taking all the formal con 
stituents to be constant, and replacing all the material 



SYMBOLISM AND FUNCTIONS 51 

constituents of the one by those of the other. But, 
when two constructs are said to exhibit the same non- 
formal function, their reduction to identity is effected 
by taking certain of the material, as well as all the 
formal, constituents to be constant, and replacing all 
the remaining material constituents of the one by those 
of the other. A formal function is a function of all the 
material constituents, since all these are to be varied ; 
but a non-formal function is a function of only some of 
the material constituents, because only some of these 
are to be varied. 

We may take the following as illustrations of formal 
functions : The construct a good boy is the same 
function of the variants good and boy as is a diffi 
cult problem of the variants difficult and problem ; 
Socrates is wise is the same function of Socrates and 
wise, as is London is populous of London and popu 
lous ; red or heavy is the same function of red and 
heavy as is loud or pleasant of loud and pleasant. 
We may compare these simple examples with similarly 
simple examples in arithmetic. The arithmetical con 
struct three days plus seven days is the same function 
of the two variants three days and seven days as is 
* five feet plus four feet of the two variants five feet 
and four feet ; four days multiplied by three is the 
same function of four days and three as is seven feet 
multiplied by two of seven feet and two, etc. These 
illustrate formal functions because the only constituents 
which are constant are formal: namely a, is, or/ 
plus/ * multiplied by/ respectively. Each of the above 
examples exhibits a specific formal function, and serves 
to explain the general notion of a formal function. We 

42 



52 CHAPTER III 

may take similar examples to illustrate the general 
notion of a non-formal function. Thus taking boy as 
constant, a good boy is the same function of good as 
is a tall boy of tall; taking good as constant, a good 
boy is the same function of boy as is a good action 
of action \ taking pleasant as constant, loud or pleasant ^ 
is the same function of loud as is bright or pleasant 
of bright ; taking wise as constant, Socrates is wise 
is the same function of Socrates as is Plato is wise 
of Plato ; taking Socrates as constant, Socrates is 
wise is the same function of wise as is Socrates is 
poor of poor, etc., etc. And in general the specific 
function exhibited by a given construct varies according 
to the constituents of the construct that operate as 
variants 1 . 



1 It will be observed that in the above illustrations of non-formal 
functions we have used adjectives and substantives indifferently as 
constants or as variables. Now in Mr Russell s first introduction of 
the notion of function, he appears to limit the application of the 
notion to the case where the substantive is variable and the adjective 
is constant. It is true that he extends the notion to include the cases 
in which the reverse holds; yet throughout he adopts an absolute 
distinction between the two constituents of a proposition which I 
have called substantive and adjective, inasmuch as he treats the sub 
stantive as the typical kind of entity which can stand by itself, the 
adjective never being allowed to stand by itself. Thus I am repeating 
his illustration in giving Socrates is wise as the same function of 
Socrates as is Plato is wise of Plato, since here the substantive terms 
Socrates and Plato are allowed to stand by themselves. But the 
parallel example, that Socrates is wise is the same function of wise 
as is Socrates is poor vlpoor, is not recognised by Mr Russell, be 
cause he does not allow such adjective-terms as wise and poor to 
stand by themselves. The consequences of this contrast, which I hold 
to be fundamentally fallacious, between the substantive and the 
adjective as constituents of a proposition, infect the whole of his logical 



SYMBOLISM AND FUNCTIONS 53 

6. A classification has been given, in an earlier 
section, of those formal constituents of a construct that 
are expressible in words or in shorthand symbols under 
stood as equivalent to words. Such formal constituents 
may be called explicit in distinction from others which 
are more or less latent and not usually expressed in 
words. Reserving the name l constituent, for the 
material variants, and formal component for those 
formal constants that are explicitly expressed, the 
implicit formal constants may be conveniently termed 
elements of form. Of these, several different kinds 
are to be distinguished : 

1 i ) Ties. These are more or less latent elements of 
form, inasmuch as it is a matter of accident whether they 
are expressed by some separate word or by some form 
of grammatical inflection. 

(2) Brackets. A construct may be composed of 
sub-constructs, and these again of sub-sub-constructs 

system. Without entering into elaborate detail, it would be impossible 
fully to justify my difference from Mr Russell on this matter; but 
what I take to be perhaps the root of the error is that he treats the 
general notion of function before giving examples of the simplest 
functional forms upon which the more complicated functions are built. 
It is true that he illustrates a function by such an elementary example 
as x is a man where x stands indifferently for Socrates or Plato, etc., 
but he does not bring out the speciality of this form of proposition, 
which does in fact exhibit the specific function which is constructed 
by means of the copula is. In mathematics the general notion of 
function is reached by building up constructs out of such elementary 
functions as those indicated by + x etc., but in Mr Russell s 
system it seems impossible to explain and reduce to systematic 
symbolisation the process by which any prepositional function what 
ever is constructed. 

I hope to treat more fully elsewhere this point of difference be 
tween Mr Russell s system and my own. 



54 CHAPTER III 

and so on, until we reach the ultimate constituents, 
namely those that are expressed, not as constructs, but 
as simples/ where by simple is not meant incapable 
of analysis, but merely unanalysed. The operation of 
binding constituents into a unity to constitute a sub- 
construct I shall call bracketing. In speaking, the dis 
tribution of brackets is indicated by pauses or vocal 
inflections; and, in writing, by punctuation marks. 
But, as the employment of these signs is not governed 
by any systematic principle, they must be replaced in 
logical or mathematical symbolism by some conven 
tional notation. 

(3) Connectedness. Two sub- constructs will be called 
unconnected when one is a function of the simple terms 
a, b, c (say), and the other of the simple terms d, e 
the terms of the one not recurring in the other. On the 
other hand, two sub-constructs will be called connected 
when one is a function of a, b, c (say), and the other of 
a, e the term a recurring in the two. This distinction 
is of importance when we have to determine what con 
stituents of a function can be taken as variants ; for 
the several variants for a function must be indepen 
dently variable, and in the case of any two complex 
constituents, if these are connected (in the sense ex 
plained), they cannot be made to vary independently 
the one of the other. Thus, in the above illustration of 
two sub-constructs that are respectively functions of 
a, b, c and of a, e, the variants for the function exhibited 
by the construct must be taken to be the simple 
constituents a, b, c, e, and not the connected sub- 
constructs themselves. But, when a construct contains 
unconnected sub-constructs, as in the example of the 



SYMBOLISM AND FUNCTIONS 55 

sub-constructs that are functions respectively of a, b, c 
and of d, e, then it may be regarded either as a function of 
the several ultimate terms involved in the different sub- 
constructs, namely a, b, c, d, e ; or alternatively, as a 
function of the sub-constructs themselves. 

(4) Categories. Every material, and therefore vari 
able, constituent belongs to a specific logical category 
or sub-category which is not usually expressed in words. 
Thus the proposition Socrates is wise is understood 
as it stands without being expanded into the form 
The substantive Socrates is characterised by the 
adjective wise/ Nevertheless the formal significance 
of the proposition for the thinker depends upon his 
conceiving of * Socrates as belonging to the category 
substantive, and of wise as belonging to the category 
adjective. These must therefore be included amongst 
the latent elements of form. It further follows from 
the recognition of this formal element, latent in every 
material constituent, that the range of variation for 
any material constituent is determined by the logical 
category substantive, adjective, relational adjective, 
as the case may be to which it belongs. In other 
words, the material constituents which may replace one 
another, in order that the construct may exhibit the 
same function in its varied exemplifications, must all 
belong to the same logical category or sub-category. 

7. This account of the formal elements of a con 
struct leads to an examination of different types of 
function. Amongst the functions of logic the con 
junctional and the predicational are the most funda 
mental. A function is called conjunctional when the 
component that determines its form is the negative not 



5 6 CHAPTER III 

or some logical conjunction ; and the variants for such 
a function are always, strictly speaking, propositions, 
as is also the construct itself. A function is called 
predicational when the component that determines its 
form is the characterising tie, which unites two variants 
related to one another as substantive to adjective. Thus 
there is only one elementary predicational function, 
namely the characterising function represented by the 
copula is ; whereas there are five elementary conjunc 
tional functions represented respectively by the opera 
tors, not/ and, if and its converse, or/ not-both. 
Just as a conjunctional function may exhibit any degree 
of complexity made up of these elementary conjunctional 
functions, so a predicational function may exhibit any 
degree of complexity made up of recurrences of the 
characterising function in sub-constructs and sub-sub- 
constructs, etc. An important distinction between these 
two types of function introduces the notion of func 
tional homogeneity. A function is said to be homo 
geneous when all its variants belong to the same 
category as itself. Now, since a conjunctional function 
takes propositions as its variants and is itself a pro 
position, it illustrates a homogeneous function ; but, 
since a predicational function constitutes a construct 
under the category proposition out of constituents under 
the respective categories, substantive and adjective, it 
illustrates a heterogeneous function. Under this head 
are also to be included secondary propositions which 
predicate adjectives of primary propositions, and pro 
positions which predicate secondary adjectives of pri 
mary adjectives; for the subjects of these propositions 
are quasi-substantives, and the propositions themselves 



SYMBOLISM AND FUNCTIONS 57 

are of a different order of category from their con 
stituent terms. 

8. We will proceed to apply the notion of con 
nectedness to these two types of function. A conjunc 
tional function is a function of those prepositional sub- 
constructs which are unconnected, but not of those 
which are connected with one another through identity 
of some of the terms involved. For such sub-constructs, 
though properly regarded as constituents, cannot be 
taken as variants, since they cannot be freely varied 
independently of one another. Thus the variants for a 
conjunctional function which is also connectional are 
not the connected sub-constructs themselves, but the 
ultimate propositions or simples of which they are 
constituted; e.g. in the construct 

{(p and q] or (p and r}} and (x or y) 

the constituents that may be taken as variants are 
/, q, r, (x orj); and in the construct 

{(p and q} or (/ and r)} and (q or y) 

the only constituents that can be taken as variants are 
A #> r > y> I n these symbolic illustrations, the ultimate 
constituents are unanalysed propositions ; but the same 
distinction between connected and unconnected sub- 
constructs holds for a conjunctional function of pro 
positions that are expressed analytically in terms of 
subject and predicate. For example, 1 A is/ or B is q 
illustrates a conjunctional function of the two uncon 
nected sub-constructs * A is/, B is q On the other 
hand, A is p or A is q is not a function of the sub- 
constructs A is p* A is q because these are con 
nected; but must be taken as a function of the three 



5 8 CHAPTER III 

ultimate constituents A, p, q. Again, A is/ or B is/ 
is not a function of A is/, is/, but of the ultimate 
constituents A, B, p. The connectedness in the former 
case is through identity of the substantive A ; in the 
latter through identity of the adjective /. Similar 
examples of connectedness occur, in which if or not- 
both or and enter in the place of or. 

Ordinary language adopts abbreviated expressions 
for propositions that are connected, through identity of 
subject, by constructing a compound predicate, e.g. A 
is/ or q, A is/ and q ; as also for propositions that 
are connected, through identity of predicate, by con 
structing a compound subject, e.g. A or B is /, A 
and B are/. This is extended to any number of terms 
enumeratively assigned for which language supplies us 
with a special condensed mode of expression. Thus the 
alternative function is condensed into the form : Some 
one or other of the enumerated items is / ; and the 
conjunctive function into the form: Every one of the 
enumerated items is/. Such forms are usually restricted 
to enumerations of substantival items : for example, 
Some one of the apostles was a traitor, Every one of 
the apostles was a Jew. But it is possible to extend 
the form to enumerations of propositional or of adjec 
tival items ; for example, Some one of the axioms of 
Euclid is unnecessary for the purpose of establishing 
the theorems of geometry ; or Every one of the 
qualities characterising A, B, C characterises D! 

9. A special notation has been adopted by the 
symbolists for representing such condensed expressions. 
In this notation, an illustrative symbol such as x enters 
as an apparent variable (to use Peano s phraseology) ; 



SYMBOLISM AND FUNCTIONS 59 

by which is meant that the proposition in which x 
occurs though it appears to be, yet is not in reality 
about x, inasmuch as its content is not changed when 
any other symbol, say y, is substituted for x. The 
typical mode of formulating propositions on this prin 
ciple is : Every item, say x, is p, or Some item, say 
x, is /, where it is obvious that the force of the pro 
position would be unaltered if we substituted s, or y t 
or 2, for x. If X is the name of the class that comprises 
all such items as x, then the above forms are equi 
valent to Every X is p? and Some X is p* respec 
tively. The ultimate constituents of such universal or 
particular propositions are the simple propositions of 
the form l x is/ which are conjunctively combined for 
the universal, and alternatively combined for the par 
ticular. The phrases * Every X, Some X? therefore, 
though obviously constituents of the sentence, do not 
denote genuine constituents of the proposition of which 
the sentence is the verbal expression. Since then the 
constituents of the general proposition are singular 
propositions of the form * x is />, such a class-name as X 
and such a variable name as x, which are in danger 
of being identified, must be carefully distinguished. To 
the former the distributives some or every can be pre 
fixed, never to the latter. [See Part I, Chapter VI L] 

When we use a symbolic variable or illustrative 
symbol x to construct the proposition x is p 1 say, 
x stands, not for a class-name, but for a special 
kind of singular name, only differing from the ordinary 
singular name in that it stands indifferently for any 
substantive name, such as Socrates or Cromwell or 
this table or yonder chair. To bring out more pre- 



60 CHAPTER III 

cisely the distinction between a symbolic variable and 
a class-name, we may suppose that in a certain context 
s stands indifferently for my person such as Socrates 
or Cromwell ; or again, indifferently for any article 
of furniture such as this table or yonder chair. 
Now person and article of furniture are class-names, 
and in the instances adduced the symbolic variable s 
stands not for the class-name but in fact for any 
singular name (proper or descriptive) that denotes an 
individual comprised in the class person for the one 
case, and the class article of furniture for the other case. 
What holds of a substantive-name s holds also of an 
adjective-name p or of a class-name c. Thus, in 
the form l s is p, where zs represents the charac 
terising tie, p stands for any one indifferently assign 
able adjective comprised, say, in the class colour, but 
not for the class itself to which the distributives 
every or some can be prefixed. Again, in the 
form s is comprised in c, 1 c represents a singular 
class-name standing for any one indifferently assignable 
class; and the limits of variation for the variable c 
could be expressed in terms of a class of a higher 
order comprising it. Thus the symbol c is equivalent to 
a variable proper class-name, and, like the substantive- 
name s and the adjective-name p, is to be contrasted 
with the class in which it is comprised. The names 
substantive, adjective, proposition, etc., which denote 
logical categories, i.e. the ultimate comprising classes, 
are not variable proper names, but names bearing fixed 
or constant significance, having so far the character of 
shorthand symbols in that they stand for logical con 
stants, not for material variables. Thus the employ- 



SYMBOLISM AND FUNCTIONS 61 

ment of the illustrative symbol as an apparent variable 
i.e. to stand indifferently for any one or another 
object makes possible the use of the same symbol, 
recurring in a given context, to stand for the same 
object. It thus fulfils the same function in a complex 
symbolic formula as the proper name in ordinary narra 
tive, where the use of the pronoun in complicated cases 
would be ambiguous. The construction of such formulae 
requires the use, in a symbolic system, of apparent vari 
ables in place of class-names. 

10. We have seen that certain phrases containing, 
implicitly or explicitly, the conjunctions and or or, though 
linguistically intelligible, do not really represent genuine 
constructs. This raises a wider and more fundamental 
problem in regard to the nature of logical conjunctions 
when used in constructingacompound out of enumerated 
items. Can conjunctions serve to construct compound 
substantives or compound adjectives in the same way 
as they operate in constructing compound propositions ? 
Now I shall maintain that while the nature of an adjec 
tive is such that we may properly construct a compound 
adjective out of simple adjectives just as we may 
construct a compound proposition out of simple pro 
positions, yet the nature of any term functioning as a 
substantive is such that it is impossible to construct a 
genuine compound substantive. Thus rational and 
animated represents a genuine conjunctive adjective, 
since it is equivalent in meaning to the simple adjective 
human ; and one or other of the colours approximat 
ing to red is a genuine alternative adjective, since it is 
equivalent in meaning to the simple adjective reddish. 
And again, more generally, where no single adjectival 



62 CHAPTER III 

word represents such a conjunction of adjectives as 
square and heavy, red or green, these are still to be 
regarded as genuine adjectival constructs on the ground 
that they agree in all essentially logical respects with 
simple adjectives, from which in fact they cannot be 
distinguished by any universal criterion. It follows, 
therefore, that no contradiction will ensue from replacing 
a simple by a compound adjective in any general for 
mula holding of all adjectives as such. At the same 
time it must be pointed out, as regards alternative adjec 
tival constructs, that no single or determinate adjective 
can be identified with such an alternative or indetermi 
nate adjective as red or green, one or other of the 
colours approximating to red. In this respect, as we 
shall see, an alternative adjectival construct precisely 
resembles a substantival construct. Turning then to 
substantival constructs, it is obvious in the first place 
that a conjunctive enumeration of substantives such as 
Peter and James or Every one of the apostles does not 
represent any single or determinate man. 1 1 might, how 
ever, be maintained that such phrases represent a couple 
of men or a class of men, and that a couple or a class 
comprising substantives is itself of the nature of a sub 
stantive. Such a view would, however, involve a con 
fusion between the enumerative and the conjunctional 
and. A statement about Peter and James or Every 
one of the apostles is really not about the compound 
construct that appears to be denoted by its subject- 
term, but must be analysed into a conjunctive compound 
of singular propositions. Thus in the statement Peter 
and James were fishermen the subject-term uses and 
enumeratively. The conjunctional and can be shown 



SYMBOLISM AND FUNCTIONS 63 

to enter only when we analyse the statement into the 
form Peter was a fisherman and James was a fisher 
man. The case of an alternative enumeration of sub 
stantives, such as * Peter or James or Some one of the 
apostles, is less obvious than that of a conjunctive 
enumeration of substantives. To prove that the alter 
native enumeration does not represent a genuine sub 
stantive, it will be convenient to take a proposition in 
which the enumeration occurs in the predicate. Thus 
f Nathaniel is one of the apostles or Bartholomew is 
one of the apostles would appear to be expressible in 
the form Nathaniel is-identical-with one or other of 
the apostles or Bartholomew is-identical-with one or 
other of the apostles. But, if this is allowed, the con 
junction of these two propositions would imply that 
Nathaniel is-identical-with Bartholomew/ since things 
that are identical with the same thing are identical with 
one another. Now that Nathaniel is-identical-with 
Bartholomew may or may not be the case; but it cer 
tainly would not follow from the fact that Nathaniel 
was one of the apostles and that Bartholomew was one 
of the apostles. I n order correctly to formulate the pro 
position Nathaniel was one of the apostles in terms of 
the relation of identity, it must be rendered: Nathaniel 
is-identical-with Peter or identical-with Bartholomew or 
identical-with Thaddeus, etc. In this form, the alter 
nants are not the proper or substantival names Peter, 
Bartholomew, Thaddeus, etc., but the adjectival terms 
identical with Peter, identical with Bartholomew, 
identical with Thaddeus, etc. These latter being re 
cognised as adjectives, the reconstructed proposition 
assumes the form A is / or q or r, etc. where is 



6 4 CHAPTER III 

represents the characterising tie, and/, q, r... stand 
for adjectives, so that (as alleged above) the new 
predicate expresses a genuine construct. 

ii. A further and more general explanation may 
now be given of the principle according to which a 
proposition containing a fictitious construct must be re 
formulated. What holds of the relation of identity (as 
in the particular example concerning the apostles) holds 
of any relation whatever : that is to say, taking r to 
stand for any relation, the phrase ? to a or b or c... 
does not express a genuine construct and must be re 
placed by the phrase r to a or r to b or r to c... which 
is an alternative of adjectives. For example, the pro 
position This action will injure either Germany or 
England must be transformed into This action will 
either injure Germany or injure England. The essen 
tial points in this transformation can best be indicated 
with the help of vertical lines for brackets. Thus : 

x | is r to | a or b or c 
is corrected into 

x\ is | r to a or r to b or r to c. 

In the former the two principal constituents of the pro 
position are linked by the relational predication is r 
to, in the latter by the characterising tie is. In order 
that the predicate in the latter case should constitute a 
genuine construct, what is essential is, not that the 
subject term should stand for a substantive in any abso 
lute sense, but only that it should function as a sub 
stantive relatively to the adjectival predicate; and it is 
the characterising tie which indicates this relative con 
ception of substantive to adjective. Thus the term x 



SYMBOLISM AND FUNCTIONS 65 

may be either a substantive proper, an adjective or a 
proposition, and the same holds of the terms a, b, c> 
with which x is connected by the relation r. 

Examples may be given of propositions based upon 
the forms *x is r to a or b? ^x is r to a and 3, in order 
further to illustrate the principles under discussion. 

(1) In the example just given: This action will 
injure either Germany or England, which must be 
rendered This action will either injure England or in 
jure Germany/ the terms x, a, 6, are all substantives 
proper. But taking 

(2) p characterises either a or b or , which has to 
be transformed into p either characterises a or charac 
terises b or characterises c* the subject term is an ad^ 
jective and may be called primary relatively to the 
predicate terms which function as secondary adjectives 1 . 

In (3): A has asserted/ or^ orr, y the subject term 
stands for a person (i.e. for a substantive proper), and 
the terms/, q, r in the predicate are propositions. Since 
here the terms alternatively combined are themselves 
propositions, the expression as it stands would be correct 
if its intention were to state that the compound propo 
sition p or q or r was asserted by A. But, if it were 
intended to state that one or other of the assertions 
p, q, r had been made by A 9 then (3) should be amended 

1 The predication characterises, like injures in the previous example, 
is expressed by a verb ; but, as explained in Part I, Chapter XIII, 
section 5, any verb may be resolved into an adjective or relation 
preceded by the characterising tie. Thus, in order to show more 
explicitly that the principal constituents are united by the characterising 
tie, proposition (2) should be expanded into the form: / is charac 
terised as either characterising a or characterising b or characterising 
c. Similarly for other examples. 

j. L. ii 5 



66 CHAPTER III 

(as in the preceding examples) into the form: A has 
asserted/ or has asserted q or has asserted r! 

(4) The proposition : *g is characterised by all the 
adjectives that characterise a and b and c exhibits a 
higher degree of complexity than those previously given 
since it introduces the two correlatives characterising 
and characterised by. It illustrates a type of proposi 
tion which plays an important part in the theory of 
induction; and is a specific case of the more general 
form: *g is r to everything that is r to a and b and c* 
As thus formulated it contains the fictitious conjunctive 
construct a and b and c? where a, b, c function as 
substantives. To eliminate this fictitious construct, the 
statement must be reformulated thus : l g is character 
ised by every adjective that characterises a and charac 
terises b and characterises c! But there still remains 
the fictitious construct prefaced by the distributive 
phrase every adjective/ The final correction must be 
made by introducing an apparent variable as was re 
quired in reformulating the elementary forms of pro 
position: Every M is p Some M is p! Thus: 
Every adjective, say;r, that characterises a and charac 
terises b and characterises c also characterises g! 

12. The above exposition of functions is funda 
mentally opposed to that given in the Principia Mathe- 
matica. The first point of difference to be emphasised 
concerns Mr Russell s view of the relation between 
what he calls a prepositional function, and function 
in the sense in which it is universally understood in 
mathematics. The latter he terms a descriptive function, 
and maintains that it is derivable from the nature of the 
propositional function; whereas it appears to me that 



SYMBOLISM AND FUNCTIONS 67 

the reverse is the case, and that his prepositional func 
tion is nothing but a particular case of the mathematical 
function. The general nature of a descriptive function 
can be illustrated by taking a proposition say about 
The teacher of y! This phrase illustrates what is meant 
by a descriptive function, the full meaning of which can 
be indicated only by showing how it may enter into a 
proposition such as (a) : The teacher of y was a Scotch 
man. Now we may agree with Mr Russell that this 
proposition could not be interpreted as true, unless y 
had one and only one teacher. On this interpretation 
the full force of the proposition is explicated as follows : 
(a) There is a being, say b, of which the following 
statements may be made: 

(1) that b was a Scotchman; 

(2) that b taught y\ 

(3) that no being other than b taught y. 

This analysis in which the describing relation is teaching 
is typical of all cases in which a descriptive function is 
used in a proposition. To illustrate a mathematical func 
tion ofjy, for teacher-of subs \\\x\te greater- by-$-than\ so 
that y + 3 stands for the quantity that is greater by 3 
than y! Again for the predication is-a-Scotchman substi 
tute is-divisible-by-Af. Thus, in place of the proposition 
The teacher of y was a Scotchman/ we have con 
structed the proposition (6): y + 3 is divisible by 4, the 
full force of which is rendered as follows : 

(6) There is a quantity, say b, of which the fol 
lowing statements may be made: 



(1) that b is divisible by 4 ; 

(2) that b is greater-by-3 th 

(3) that no quantity other than b is greater-by-3- 



52 



68 CHAPTER III 

Thirdly, to illustrate a prepositional function, for 
divisible-by-4 substitute the predicate dubious , for the 
quantitative construct y + 3 substitute the prepositional 
construct y is /. We have thus constructed the se 
condary proposition (c): That y is/ is dubious/ of 
which the full force is rendered as follows : 

(c) There is a proposition, say b, about y, of which 
the following statements may be made: 

(1) that b is dubious; 

(2) that b predicates-/-aboutj/; 

(3) that no proposition other than b predicates- 
/-about y. 

Now in example (a) the ground for asserting unique 
ness of the construct the teacher ofy is merely empirical 
or factual; but in example (b] the necessary and sufficient 
condition for the uniqueness of the construct y + 3 is its 
mathematical form, as indicated by the symbol + ; and 
in example (^) the uniqueness of the corresponding con 
struct y is p similarly depends upon its logical form, as 
indicated by the logical constant is. Dismissing the em 
pirical example which requires no further discussion, it 
must be pointed out as regards the quantitative function 
(b) and the prepositional function (c) that these illustrate 
not quantitative or propositional functions in general 
but certain specific functions : in the former case that 
which is constructed by means of plus, and in the latter 
case, that which is constructed by means of is. The 
former may be called the additive and the latter the 
characterising function. Just as the quantitative con 
struct y + a would not yield a quantity unless y and a 
were themselves quantities of the same kind ; so the 
propositional construct y is p would not yield a pro 
position unless the two constituents y and p were, in 



SYMBOLISM AND FUNCTIONS 69 

their nature, relatively to one another as substantive to 
adjective. A specifically different form of quantitative 
construct would have been obtained if fory + a we had 
substituted y : a. Similarly a specifically different pro- 
positional form of construct would have been obtained 
if for s is p we had substituted % is identical with y. 
In both cases the uniqueness of the construct is secured 
by the nature of the operator involved ; viz., + which 
yields a sum, or : which yields a ratio for the two quanti 
tative constructs ; and is and is identical with for the 
two prepositional constructs. If there is any difference 
between the uniqueness of the prepositional construct 
when its constituents are given and that of the mathe 
matical construct when its constituents are given, it is 
that the uniqueness in the former case is assumed on 
the ground of its intuitive evidence realised in the mental 
act of constructing the proposition, whereas in the latter 
the uniqueness may require and maybe capable of formal 
demonstration. 

1 3. Before continuing the discussion of my differ 
ences from Mr Russell, I shall examine more precisely 
what he means by a descriptive function. A descriptive 
function (p. 245) is defined to be a phrase of the form : 
the term x that has the relation r to the term y! In 
this definition the sole emphasis is to be laid on the 
predesignation the. Now, just as we speak of the 
quantity .?+/, so we speak of the proposition s is p! 
But these quantitative and prepositional phrases differ 
from ordinary descriptive phrases such as the writer 
of Waver ley or the teacher of Xenophon which are 
of the general form : the thing x that is r to the thing 
y in that they do not explicitly contain any descriptive 



7 o CHAPTER III 

relation r (writing or teaching). The arithmetical form 
<s+/ and the prepositional form s is / having in 
common this negative characteristic, I shall proceed to 
maintain that they are, in all essential logical respects, 
identical in nature ; and, if either of the two can be 
explicated into the form of a descriptive function, so 
can the other. We may attempt to express these forms 
explicitly as descriptive functions by introducing, as 
the describing relation, constructed by. Thus the pro- 
positional function maybe rendered: the proposition 
x constructed by means of is out of the constituents s 
and/ ; and the quantitative function may be rendered : 
the quantity x constructed by means of plus out of the 
constituents s and /. This attempt reduces the state 
ment of equivalence of the construct with the proposed 
descriptive phrase to a mere tautology ; for the pro 
position x constructed by means of is out of the con 
stituents s and / is merely a lengthened expression 
for the proposition s is/ ; and similarly the quantity 
x constructed by means of plus out of the constituents 
s and / is merely a lengthened expression for the 
quantity s+p! It thus turns out that the x thus 
introduced in the completely formulated descriptive 
phrase stands merely for the function itself, i.e. in the 
one case for s is/ and in the other for .$+/. Follow 
ing Russell in his demand that a descriptive function 
must only be defined in use, the statement that s is 
/ is dubious or that s+p is divisible by 4 must be 
rendered the proposition x constructed etc. (as above) 
is dubious, or the quantity ^constructed etc. (as above) 
is divisible by 4. In this way the original statements: 
the proposition s is / is dubious/ and the quantity 



SYMBOLISM AND FUNCTIONS 71 

s+p is divisible by 4, which were supposed to require 
definition, are after all defined tautologically. 

14. Another way of attacking Russell s preposi 
tional function, which in fact presents only another 
aspect of the same criticism, is to ask: What are the 
variants for any given proposional function, and what 
function is it that a given propositional form exhibits ? 
In his first introduction of the notion of propositional 
function, Mr Russell gives three quite different appli 
cations of the symbol for a function. According to his 
first definition, $x is called a propositional function when 
x is variable provided that when x is replaced by the 
constant a, $a represents a proposition. Now here the 
symbol for a function is first used along with a variable 
and then along with a constant ; although Russell insists 
that <^a is not a function but a proposition, and that $x 
is not a proposition but a function. It seems to me that 
he cannot attach the symbol for a function exclusively 
to a variable in this way without contradiction at every 
point; and it is for this reason that, in my account of 
functions, I have used the word variant to include both 
Russell s variable and his constant. There is yet a third 
application of the symbol for a function deliberately 
introduced in the very first paragraph of his exposition, 
by way of correcting his initial definition of propositional 
function. For his first account is that $x is to be called 
a propositional function, owing to the ambiguity or as 
I should prefer to say indeterminateness of the symbol 
x, and that it is not itself a proposition, and would only 
become a proposition when a is substituted for x. This 
is corrected, however, when he takes the example x 
is hurt which he says illustrates, not a propositional 



72 CHAPTER III 

function, but an ambiguous (i.e. an indeterminate) value 
of a prepositional function. Thus, as I have pointed 
out, he illustrates the use of the word function in his 
first paragraph in three different ways which are sym 
bolised as follows : a is hurt, x is hurt, and x is hurt. 
The last application of the word function is that which 
he wishes to be finally adopted; but, in spite of this, he 
continually uses the word in both of the two other ap 
plications. It is still more surprising that, on page 6 of 
his Introduction, where he gives a preliminary account 
of the ideas and notations of logical symbolism, he uses 
the word function without any explanation of its meaning, 
and in deliberate defiance of his own later definition. 
Thus he speaks of the fundamental functions of pro 
positions in these words : an aggregation of propositions 
considered as wholes, not necessarily unambiguously 
determined, into a single proposition more complex than 
its constituents, is a function with propositions as argu 
ments. This account appears clearly to suggest that un 
ambiguously determined constituents are allowable as 
arguments for a function, which contradicts his explicit 
definition. He proceeds to enumerate the four funda 
mental functions of propositions which are of logical im 
portance, viz. (i) the contradictory function, which I 
have called the negative function; (2) the logical sum 
or disjunctive function, which I have called the alter 
native function; (3) the logical product, which both he 
and I call the conjunctive function; (4) the implicative 
function, for which I have used the same term. These 
four functions I have called conjunctional functions, in 
contrast to the one fundamental predicational function. 
The recognition of this distinction, which does not appear 



SYMBOLISM AND FUNCTIONS 73 

in Mr Russell s account, would have simplified and 
corrected his theory of types. But, in thus introducing 
the specific conjunctional functions, he inevitably adopts 
the familiar meaning of the mathematical term function/ 
the essence of which lies, not in the indeterminateness 
of the constituent terms, but in the identity of form that 
is exhibited in the process of substituting indifferently 
any one term for any other. 

That he is not only in disagreement with universal 
usage, but also logically mistaken, when he says that 
it is a function of which the essential characteristic is 
ambiguity and thus that $x ambiguously denotes fa, 
<f)&, fa, where fa, <f>6, $c are the various values of fa 
is shown by noting that the ambiguity attaching to fa 
is not due to the nature of (j> as a function, but to the 
nature of the symbol x itself; that is to say, fa am 
biguously denotes fa, <j>b, fa, etc., only because x am 
biguously denotes a, b, c, etc. In short a prepositional 
function has ambiguous denotation, if it contains a term 
having ambiguous denotation; whereas a prepositional 
function has unambiguous denotation, if it contains no 
term having ambiguous denotation. 

15. Hitherto, in illustrating Russell s account, we 
have taken the prepositional function to be a function 
of a single variable, viz., of the symbol for the subject 
of the proposition, the predicate standing for a constant. 
It is obvious, however, that no proposition can be re 
garded as a function of a single variant unless the pro 
position is represented by a simple letter; and we will 
therefore take the specific propositional form x is/ to 
illustrate a function of two variables. The variants of 
which this is a function would naturally be taken as the 



74 CHAPTER III 

symbols x and p themselves ; but, since Russell refuses 
to allow a predicate or adjective to stand by itself, he 
takes as the two variables the subject term x together 
with the symbolic variable x is p. The symbolic ex 
pression x is/ may be read .ar-blank is / ; by which 
is meant that instead of the full prepositional form x 
is /, we suppose that the subject-term x is omitted, 
leaving a blank. But, if we use a blank symbol for the 
subject-term, we ought in consistency to be allowed to 
use a similar blank symbol for the predicate term. This 
would give rise to nine combinations all of which are of 
the same prepositional form: this is hurt, x is hurt, 
this is// is hurt, this is/, is/, x is/, x is/, 
and finally x is/. Of these nine phrases, Russell uses 
only this is hurt, x is hurt and *fc is hurt ; of which 
the two latter illustrate the two admittedly different 
meanings or applications of the general notion foe, i.e. 
of the prepositional function. Now, though his first 
reference is to a prepositional function taking a single 
argument, nevertheless he allows that any proposition 
(as distinguished from a prepositional function) when 
analysed contains at least two constituents. For example, 
the proposition this is hurt as analysed contains the 
two constituents this and f x is hurt. In my view, 
there is no ground whatever for preferring this analysis 
either to that in which the constituents are hurt and 
this is/, or to that in which the constituents are this 
is/ and l x is hurt. But, returning to his own analysis 
in which this and x is hurt are assigned as the two 
constituents of this is hurt, as also x and *x is/ 
as the two constituents of x is / ; we must insist 
upon asking: What is the specific function for the 



SYMBOLISM AND FUNCTIONS 75 

case of the proposition l x is / when its two arguments 
are taken to be x and *x is/ ? Mr Russell only tells 
us that *x is/ = /(#, x is/) where the specific symbol 
/"has nowhere been defined by him. It is as if he had 
said that the quantitative function #+/ has for its 
two constituents, variants or arguments: (i) x and (2) 
x+p. Now according to this analysis of the nature of 
a function, the process by which a function is constructed 
out of two variables is to substitute in one of these 
variables x for , so that taking a similar example to 
the above, the constituents of the quantitative construct 
x~p would be x and x+p. Every mathematician 
would take as the two constituents of the construct 
x-p the two simple symbols x andfi; as Russell himself 
does in his preliminary account of the alternative func 
tion x or/, of which the two constituents are the simple 
symbols x and /. In fact he can only take a function 
of a single variable as ambiguously denoting a pro 
position, by starting with what I have called a non- 
formal function, e.g. *x is hurt as a non-formal function 
of^r; instead of starting with the essentially logical notion 
of a function, which is synonymous with the form of a 
construct such as l x is/ where instead of one material 
or variable constituent there are two. In short the form 
of a proposition, if it has form at all and is not simply 
expressed by a simple symbol, must contain two inde 
pendent constituents. When Mr Russell says that 
<j) (x) is a propositional function, provided that < (a) is 
a proposition, he provides us with no indication as to 
the form that < (a) must assume in order that < (a) 
shall constitute a proposition. 



CHAPTER IV 

THE CATEGORICAL SYLLOGISM 

i. As the relation between implication and in 
ference has already been explained, we may treat the 
syllogism indifferently as a species either of implication 
or of inference: regarded as implication, the propositions 
concerned must be spoken of as implicants and impli 
cate; regarded as inference, we speak of them as pre 
misses and conclusion. The term syllogism is strictly 
confined to one only of the many forms of demonstrative 
inference; and in this strict usage must be defined as 
an argument containing two premisses and a conclusion, 
involving between them three terms, each of which 
occurs in two different propositions. That occurring as 
predicate in the conclusion is called the major term; 
that occurring as subject in the conclusion, the minor 
term; and that not occurring in the conclusion, the 
middle term. The distinction between the major and 
minor terms determines which of the premisses shall be 
called major and which minor: that which contains 
the predicate of the conclusion being called the major 
premiss; and that which contains the subject of the 
conclusion being called the minor premiss. Reference 
to the conclusion is thus required before the premisses 
can be distinguished as major or minor. The canonical 
order of the three propositions, viz. major premiss, 
minor premiss, conclusion, is purely artificial, and 
adopted only for general purposes of reference. The 
mood of a syllogism is defined by the forms (A, E, /, 



THE CATEGORICAL SYLLOGISM 77 

or O) of the three propositions constituting the major 
premiss, minor premiss, and conclusion, in their canoni 
cal order. Furthermore syllogisms are distinguished 
according to figure : the first figure being that in which 
the middle term occurs as subject in the major premiss 
and predicate in the minor ; the fourth figure being that 
in which the middle term occurs as predicate in the 
major and as subject in the minor: the second figure, 
that in which the middle term occurs as predicate in 
both premisses; and the third figure that in which the 
middle term occurs as subject in both premisses. Two 
syllogisms would be said to be of different form, although 
they might agree in mood, provided they differed in 
figure. 

2. There are two opposite tendencies in the choice 
of illustrations of the syllogism, both of which, in my 
view, should be avoided. The first is to select examples 
composed of propositions, each of which is universally 
accepted as true. But such illustrations hinder the 
learner from examining the validity of the inferential 
process from premisses to conclusion, since he is apt to 
assume validity because of his familiarity with the pro 
positions as being generally accepted. The opposite 
course, which we find amusingly illustrated by Lewis 
Carroll, is to select propositions which are obviously 
false. But this leads the learner to regard the syllogism 
merely as a kind of game, and as having no real signi 
ficance in actual thought procedure. It is preferable, 
therefore, to select propositions which are dubious, 
or which are affirmed by some persons and denied 
by others. Of such propositions important kinds are 
(i) those which deal with political, ethical, or similar 



78 CHAPTER IV 

topics in general, e.g. Lying is sometimes right, All 
countries that adopt free- trade are prosperous, The 
suffrage should not be extended to uneducated persons ; 
(2) those which exercise the faculty of judgment, in the 
Kantian sense, upon some individual case, e.g. This 
man is untrustworthy, The Niche is finer than the 
Venus of Milo/ Esau is a more lovable person than 
Jacob. 

3. Correlative to the syllogism we may here in 
troduce the antilogism, in reference to which the above 
principle of selecting examples will be seen to have 
special significance. An antilogism may be defined as a 
formal disjunction of two, three, or more propositions, 
each of which is entertained hypothetically. When 
limited to three propositions constituting a disjunctive 
trio, the antilogism may be formulated in terms of illus 
trative symbols as follows: the three propositions P, Q, 
and R cannot be true together. It is then seen that 
just as the disjunction of P and Q is equivalent to the 
implication If P is true, then Q is false, so the disjunc 
tion of P y Q, and R is equivalent to each of the three 
implications : 

(1) If P and Q are true, then R is false, 

(2) If P and R are true, then Q is false, 

(3) If R and Q are true, then P is false. 

We may put forward the following example of an 
antilogism, no one of the propositions of which would 
be universally acknowledged either as true or as false, 
but which taken together are formally incompatible : 

P. All tactful persons sometimes lie. 
Q. Lord Grey is a tactful person. 
R. Lord Grey never lies. 



THE CATEGORICAL SYLLOGISM 79 

Something 1 could be said in support of, as well as in 
opposition to, each of these three propositions; but it is 
obvious that they are together incompatible, and hence 
constitute an antilogism or disjunctive trio. This anti- 
logism is equivalent to each of the three following 
syllogistic implications : 

ist if All tactful persons sometimes lie 
and Lord Grey is a tactful person, 
then Lord Grey sometimes lies. 

2nd if All tactful persons sometimes lie 
and Lord Grey never lies, 
then Lord Grey is not a tactful person. 

3rd if Lord Grey never lies 

and Lord Grey is a tactful person, 
then Some tactful persons never lie. 

4. The propositions in each of these syllogisms 
are in the canonical order of major, minor, conclusion, 
and the syllogisms will be recognised as being in the 
first, second, and third figures respectively. In defining 
the figures of syllogism we may, in fact, separate the 
first three from the fourth in that the former contain 
one and only one term standing in one proposition as 
subject and in another as predicate, while in the fourth 
figure all three terms occupy this double position. Such 
a term may be called a class-term, on the ground that 
a class-term has a partly adjectival meaning, and as such 
serves appropriately as predicate; and partly a sub 
stantival meaning, and as such serves appropriately as 
subject. The first three figures, then, containing only 
one class-term, are distinguished from one another 
according as this term occupies one or another position. 
In the first figure it serves as the middle term; in the 



8o CHAPTER IV 

second figure as the major term ; and in the third figure 
as the minor term. Taking the above antilogism as 
illustrative, we may generalise by formulating the 
following antilogistic dictum for the first three figures: 

It is impossible to conjoin together the three pro 
positions: 

Every member of a class has a certain property ; 
A certain object is included in that class ; 
This object has not that property. 

From this single antilogistic dictum we construct the 
dicta for the first three figures of syllogism, thus : 

Dictum for \st Figure 

if Every member of a class has a certain property 
and A certain object is included in that class, 
then This object must have that property. 

Dictum for 2nd Figure 

if Every member of a class has a certain property 
and A certain object has not that property, 
then This object must be excluded from the class. 

Dictum for ^rd Figure 

if A certain object has not a certain property 
and This object is included in a certain class, 
then Not every member of the class has that property. 

These dicta bring out the normal function of each of the 
first three figures in thought-process. Thus we are 
reasoning in the first figure when, having established a 
certain characteristic as belonging to every member of 
a class, we bring forward an individual object known to 
belong to the class and proceed to assert that it will 
have the characteristic common to the class. We are 
reasoning in the second figure when, having similarly 



THE CATEGORICAL SYLLOGISM 81 

established a certain characteristic as belonging to every 
member of a class, and having found that an individual 
object has not this characteristic, we proceed to assert 
that it does not belong to the class. We are reasoning 
in the third figure when we note that a certain object 
known to belong to a certain class has not a certain 
property, and proceed to assert that that property cannot 
be predicated universally of all members of the class; 
or otherwise, when, having noted that an object known 
to belong to a certain class has a certain character, we 
infer that at least one member of the class has this 
character. A peculiarity of the third figure is that it 
functions either destructively or constructively; as de 
structive, it disproves some universal proposition that 
may have been suggested ; as constructive, it naturally 
suggests the replacement of the particular conclusion 
either by a universal whose subject is restricted by some 
further adjectival characteristic, or by an unrestricted 
universal to be obtained by induction from the par 
ticular conclusion. 

5. A second illustration of an antilogism develop 
ing into three syllogisms may be chosen with the purpose 
of showing how purely formal and elementary reasoning 
underlies even the most abstract arguments. Thus : 

It is impossible to conjoin the three propositions: 

P. All possible objects of thought are such as have 
been sensationally impressed upon us; 

Q. Substance is a possible object of thought ; 

R. Substance has not been sensationally impressed 
upon us. 

Since each of these propositions has been asserted by 

j. L. ii 6 



82 CHAPTER IV 

some and denied by other philosophers, the three 
together constitute an antilogism having the same illus 
trative value as our previous example. 

Taking, first, P and Q as asserted premisses and 
not-R as conclusion, we obtain the syllogistic inference : 

P. All possible objects of thought have been sensa 
tionally impressed upon us; 

Q. Substance is a possible object of thought; 

. \ not--/?. Substance has been sensationally impressed 
upon us. 

With some explanations and modifications this syllo 
gism represents roughly one aspect of the new realistic 
philosophy. 

Taking, next, P and R as asserted premisses and 
not-Q as conclusion, we have : 

P. All possible objects of thought have been sensa 
tionally impressed upon us; 

R. Substance has not been sensationally impressed 
upon us; 

. *. not-<2. Substance is not a possible object of thought. 
This syllogism represents very fairly the position of 
Hume. 

Taking, lastly, R and Q as asserted premisses and 
not-/ 5 as conclusion, we have : 

R. Substance has not been sensationally impressed 
upon us; 

Q. Substance is a possible object of thought; 

. . not-P. Not every possible object of thought has been 
sensationally impressed upon us. 

This syllogism represents almost precisely the well- 
known position of Kant. 



THE CATEGORICAL SYLLOGISM 83 

As in our previous example these three syllogisms 
are respectively in figures i, 2, and 3; and, moreover, 
Kant s argument in figure 3 has both a destructive 
function in upsetting Hume s position; and a construc 
tive function in suggesting the replacement of the 
particular conclusion by a limited universal which would 
assign the further characteristic required for discrimi 
nating those objects of thought which have not been 
obtained by experience from those which have been 
thus obtained. 

6. Since the dicta, as formulated above, apply 
only where two of the propositions are singular or 
instantial, they must be reformulated so as to apply also 
where all the propositions are general, i.e. universal or 
particular. Furthermore, they will be adapted so as to 
determine directly all the possible variations for each 
figure. As follows: 

Dictum for Fig. i 

if Every one of a certain class C possesses (or lacks) 

a certain property P 

and Certain objects S are included in that class C, 
then These objects ,S must possess (or lack) that pro 
perty P. 

Dictum for Fig. 2 

if Every one of a certain class C possesses (or lacks) 

a certain property P 

and Certain objects 5 lack (or possess) that property P 9 
then These objects S must be excluded from the 
class C. 

Dictum for Fig. 3 

if Certain objects 5 possess (or lack) a certain pro 
perty P 

62 



8 4 CHAPTER IV 

and These objects S are included in a certain class C 
then Not every one of the class C lacks (or possesses) 

that property P. 

i.e. Some of the class C possess (or lack) that pro 
perty P. 

In each of these dicta the word objects, symbolised 
as S, represents the term that stands as subject in both 
its occurrences; the word * property P 9 the term that 
stands as predicate in both its occurrences; and the 
word class C, that term which occurs once as subject 
and again as predicate. Hence, using the symbols 
S, C, P, the first three figures are thus schematised : 

Fig. i Fig. 2 Fig. 3 

C-P C-P S-P 

S-C S-P S-C 

.-. S-P . . S-C . . C-P 

7. In order systematically to establish the moods 
which are valid in accordance with the above dicta, it 
should be noted in each figure (i) that the proposition 
S P is unrestricted as regards both quality and 
quantity; (2) that the proposition 5 C is indepen 
dently fixed in quality, but determined in quantity by 
the quantity of the unrestricted proposition ; and (3) that 
the proposition C Pis independently fixed in quantity, 
but determined in quality by the quality of the un 
restricted proposition. Thus in Fig. i, while the 
conclusion is unrestricted, the minor premiss is indepen 
dently fixed in quality but determined in quantity by 
the quantity of the conclusion; and the major premiss 
is independently fixed in quantity but determined in 
quality by the quality of the conclusion. In Fig. 2, 
while the minor premiss is unrestricted, the conclusion 



THE CATEGORICAL SYLLOGISM 85 

is independently fixed in quality but determined in 
quantity by the quantity of the minor premiss ; and the 
major premiss is independently fixed in quantity, but 
determined in quality by the quality of the minor pre 
miss. In Fig. 3, while the major premiss is unrestricted, 
the minor premiss is independently fixed in quality but 
determined in quantity by the quantity of the major 
premiss, and the conclusion is independently fixed in 
quantity but determined in quality by the quality of the 
major premiss. 

Having in the above dicta italicised the phrase in 
each case which is directly restrictive, the proposition 
which is unrestricted, i.e. may be of the form A or E 
or / or (9, is seen to be : in Fig. i , the conclusion ; in 
Fig. 2, the minor premiss ; in Fig. 3, the major premiss. 
Hence each of these figures contains four fundamental 
moods derived respectively by giving to the unrestricted 
proposition the form A, E, I or O. Besides these four 
fundamental moods there are also supernumerary moods. 
These are obtained by substituting, in the conclusion, 
a particular for a universal ; or, in the minor premiss, 
a universal for a particular; or, in the major again, a 
universal for a particular. These supernumerary moods 
will be said respectively to contain a weakened con 
clusion, a strengthened minor, or a strengthened 
major; and, in the scheme given in the next section, 
the propositions thus weakened or strengthened will 
be indicated by the raised letters w or s as the case 
may be. 

8. Adopting the method above explained, we may 
now formulate the special rules for determining the 
valid moods in each figure as follows : 



86 CHAPTER IV 

Rules for Fig. i. 

The conclusion being unrestricted in regard both to 
quality and quantity, 

(a) The major premiss must in quantity be uni 
versal, and in quality agree with the conclusion. 

(b) The minor premiss must be in quality affirma 
tive, and in quantity as wide as the conclusion. 

Rules for Fig. 2. 

The minor premiss being unrestricted in regard both 
to quality and quantity, 

(a) The major premiss must be in quantity uni 
versal, and in quality opposed to the minor. 

(b] The conclusion must be in quality negative, 
and in quantity as narrow as the minor. 

Rules for Fig. 3. 

The major premiss being unrestricted in regard both 
to quantity and quality, 

(a) The conclusion must in quantity be particular, 
and in quality agree with the major, 

(6) The minor premiss must in quality be affirma 
tive, and in quantity overlap 1 the major. 

Italicising in each case the unrestricted proposition, 
we may represent the valid moods for the first three 
figures in the following table : 

Valid Moods for the "One-Class" Figures. 



Fundamentals 



Fig. i AA^ EAE AI7 EI<9 



Supernumeraries 

sw sw 

AAI EAO 

s w s w 

Fig. 2 E^E AE E/O A6>O EAO AEO 

ss ss 

Fig. 3 ^411 10 7AI <9AO AAI EAO 

1 The minor and major will necessarily overlap if one or the other 
is universal , not otherwise. 



THE CATEGORICAL SYLLOGISM 87 

9. Having established the valid moods of the first 
three figures from a single antilogism, we proceed to 
construct those of the fourth figure also from a single 
antilogism; thus: 

Taking any three classes, it is impossible that 

The first should be wholly included in the second 
while The second is wholly excluded from the third 
and The third is partly included in the first. 

The validity of this antilogism is most naturally 
realised by representing classes as closed figures. Such 
a representation is in fact valid, although the relation 
of inclusion and exclusion of classes is not identical 
with the logical relations expressed in affirmative and 
negative propositions respectively ; for, there is a true 
analogy between the relations between classes and the 
relations between closed figures; in that the relations 
between the relations of classes are identical with the 
corresponding relations between the relations of closed 
figures. Thus adopting as the scheme of the fourth 

figure : 

C C C C C -C 

-1 V"2 *-3 *-| ^3 t- l 

the above antilogism will be thus symbolised : 

It is impossible to conjoin the following three pro 
positions : 

P. Every C T is C 2 , 
Q. NoC 2 isC 3 , 
R. Some C s is C^. 

This yields the three fundamental syllogisms 
(i) If P and Q, then not-7?; i.e. 

if Every C 1 is C 
and No C 2 is C 3 , 
then No C is C. 



88 CHAPTER IV 

(2) If Q and R, then not-/*; i.e. 

if No C 2 is C 3 
and Some C 3 is C lt 
then Not every Cj is C 2 . 

(3) If J? and P, then not-? ; i.e. 

if Some C 3 is C x 
and Every C x is C 2 , 
then Some C 2 is C 3 . 

Since the propositions of these syllogisms are 
arranged in canonical order, the valid moods in the 
fourth figure can be at once written down : AEE, EIO, 
IAL Moreover, since the conclusion of the first mood 
is universal, it may be weakened; since the minor of 
the second is particular, it may be strengthened; and 
since the major of the third is particular, it also may be 
strengthened. This yields: 

Valid Moods of the Fourth Figure. 



Fundamentals 



AEE EIO IAI 



Supernumeraries 

w s s 

AEO EAO AAI 



Here each supernumerary can only be interpreted in 
one sense, viz., as containing respectively a weakened 
conclusion, a strengthened minor, and a strengthened 
major. In contrast to this, the supernumeraries of the 
first and second figures must be interpreted as contain 
ing either a weakened conclusion or a strengthened 
minor; and those of the third figure as containing 
either a strengthened major or a strengthened minor. 

10. An antiquated prejudice has long existed 
against the inclusion of the fourth figure in logical 
doctrine, and in support of this view the ground that 
has been most frequently urged is as follows : 



THE CATEGORICAL SYLLOGISM 89 

Any argument worthy of logical recognition must 
be such as would occur in ordinary discourse. Now it 
will be found that no argument occurring in ordinary 
discourse is in the fourth figure. Hence, no argument 
in the fourth figure is worthy of logical recognition. 

This argument, being in the fourth figure, refutes 
itself; and therefore needs to be no further discussed. 

n. Having formulated certain intuitively evident 
dicta, the observance of which secures the validity of 
the syllogisms established by their means, we will pro 
ceed to formulate equally intuitive rules the violation 
of which will render syllogisms invalid. These rules 
will be found to rest upon a single fundamental con 
sideration, viz. if our data or premisses refer to some 
only of a class, no conclusion can be validly drawn 
which refers to all members of that class. This is 
technically expressed in the rule: 

(i) No term which is undistributed in its premiss 
may be distributed in the conclusion. 

This rule alone is not sufficient directly to secure 
validity, but from it we can deduce other directly 
applicable rules which, taken in conjunction with the 
first, will be sufficient to establish directly the invalidity 
of any invalid form of syllogism, In the course of 
deducing these other rules we shall make use of certain 
logical intuitions that are obvious apart from their em 
ployment in this deductive process, of which the follow 
ing may be mentioned: 

(a) that if a term is distributed in any given 
proposition, it will be undistributed in the contradictory 
proposition ; and conversely, if a term is undistributed 
in a given proposition, it will be distributed in the 



9 o CHAPTER IV 

contradictory proposition. That this is so is directly seen 
when it has been accepted on intuitive grounds that only 
universals distribute the subject term, and only nega 
tives the predicate term ; and that an A proposition is 
contradicted by an O, and an / proposition by an E. 

(b) That any syllogism can be expressed as an 
antilogism and conversely. This principle follows from 
the intuitive apprehension of the relation between im 
plication and disjunction. 

(c) That it is formally possible for any three 
terms to coincide in extension. (This particular in 
tuition is employed in the rejection of only one form of 
syllogism.) 

We are now in a position to deduce from our 
original principle, i.e. from rule (i), by means of (a), 
(b) y and (*:), other rules, the direct application of which 
will exclude any invalid forms of syllogism. 

(2) The middle term must be distributed in one or 
other of the premisses. 

To establish this, let us consider the antilogism 
which disjoins P, Q and R\ this, by (b) is equivalent 
to the syllogism If P and Q, then not-^ and also to 
the syllogism If P and R, then not-?. Taking the 
first of these, if a term X is undistributed in the premiss 
P, it must be undistributed in the conclusion not-7?, 
i.e. it must, by (a), be distributed in R. Applying this 
result to the second syllogism If P and R, then not-<2, 
we have shown that if the middle term X is undistri 
buted in the premiss P, it must be distributed in the 
premiss R. This then establishes rule (2). 

(3) If both premisses are negative, no conclusion 
can be syllogistically inferred. 



THE CATEGORICAL SYLLOGISM 91 

For, taking any two universal negative premisses, 
these can be converted (if necessary) into No P is M 
and No > is M ; which, by obversion, are respectively 
equivalent to All P is non-M and All S is non-J// 
in which the new middle term non-M is undistributed 
in both premisses. But this breaks rule (2). What 
holds of two universals will hold a fortiori if one or 
other of the two negative premisses is particular. Thus 
rule (3) is established. 

(4) A negative premiss requires a negative con 
clusion. 

For, taking again the antilogism which disjoins P, 
Q and R, this is equivalent both to the syllogism If P 
and R, then not-<2, and to the syllogism If P and Q, 
then not-R. Taking the first of these two syllogisms, 
by rule (3), if the premiss P is negative, the premiss R 
must be affirmative. Applying this result to the second 
syllogism, we have, if the premiss P is negative, the 
conclusion not-fi must be negative. This establishes 
rule (4). 

(5) A negative conclusion requires a negative 
premiss. 

This is equivalent to the statement that two affirma 
tive premisses cannot yield a negative conclusion. To 
establish this rule, we must take the several different 
figures of syllogism : 

Fig. i Fig. 2 Fig. 3 Fig. 4 

M-P P-M M-P P-M 

S-M S-M M-S M-S 

S-P S-P S-P S-P 

For the first or third figure, affirmative premisses 
with negative conclusion would entail false distribution 



9 2 CHAPTER IV 

of the major term ; which has been forbidden under our 
fundamental rule (i). Taking next the second figure, 
it would entail false distribution of the middle term, 
forbidden by rule (2). Finally taking the fourth figure, 
it would either entail some false distribution forbidden 
by rules ( i ) and (2) ; or else yield the mood A A O which 
would constitute a denial that three terms could coincide 
in extension, thus contravening (c\ This establishes 
rule (5). 

12. The five rules thus established may be re 
arranged and summed up into two rules of quality and 
two rules of distribution, viz. 

A. Rules of Quality. 

(a^ For an affirmative conclusion both premisses 
must be affirmative. 

(a^) For a negative conclusion the two premisses 
must be opposed in quality. 

B. Rules of Distribution. 

The middle term must be distributed in at 
least one of the premisses. 

No term undistributed in its premiss may be 
distributed in the conclusion. 

These rules having been framed with the purpose of 
rejecting invalid syllogisms, we may first point out that, 
irrespective of validity, there are sixty-four abstractly 
possible combinations of major, minor and conclusion. 
The Rules of Quality enable us to reject en bloc all 
moods except those coming under the following three 
heads, viz. those which contain (i) an affirmative con 
clusion (requiring affirmative major and affirmative 
minor) ; (ii) a negative major (requiring affirmative 



THE CATEGORICAL SYLLOGISM 



93 



minor and negative conclusion); (iii) a negative minor 
(requiring affirmative major and negative conclusion). 
This leads to the following table, which exhibits the 
24 possibly valid moods unrejected by the Rules of 
Quality. 





Maj. & Min. 
Univ. 


Maj. Min. 
Univ. Part. 


Maj. Min. 
Part. Univ. 


Maj. & Min. 
Part. 


Concl. Aff. 


AAI 


AAA 


All 


AIA 


IAI 


IAA 


Ill 


IIA 


Maj. Neg. 


EAO 


EAE 


EIO 


EIE 


OAO 


OAE 


OIO 


OIE 


Min. Neg. 


AEO 


AEE 


AGO 


AOE 


IEO 


IEE 


IOO 


IOE 



13. The Rules of Quality having thus been applied, 
it remains to reject such of the above 24 as violate the 
Rules of Distribution, (i) for the middle term, (ii) for 
the major term, (iii) for the minor term. This will re 
quire three special rules for each of the four figures: 

Fig. i Fig. 2 Fig. 3 Fig. 4 

M-P P-M M-P P-M 

S-M S-M M-S M-S 

S-P S-P S-P S-P 

The above scheme shows that it will be convenient 
to bracket Fig. i with Fig. 4 for the middle term, Fig. i 
with Fig. 3 for the major term, and Fig. i with Fig. 2 
for the minor term\ leading to the following: 

RULES FOR CORRECT DISTRIBUTION. 
\st of the Middle Term. 

Fig. i. If the minor is affirmative, the major must 
be universal. 

Fig. 4. If the major is affirmative, the minor must 
be universal. 



94 CHAPTER IV 

Fig. 2. One premiss must be negative; i.e. con 
clusion must be negative. 

Fig. 3. One or the other of the premisses must be 
universal. 

2nd of the Major Term. 

Figs, i and 3. If the conclusion is negative, the 
major must be negative; i.e. (in either case) the minor 
must be affirmative. 

Figs. 2 and 4. If the conclusion is negative, the 
major must be universal. 

^rd of the Minor Term. 

Figs, i and 2. If the minor is particular, the con 
clusion must be particular. 

Figs. 3 and 4. If the minor is affirmative, the con 
clusion must be particular. 

These rules have been grouped by reference to the 
term (middle, major or minor) which has to be correctly 
distributed. They will now be grouped by reference to 
the figure (ist, 2nd, 3rd or 4th) to which each applies. 
In this rearrangement we shall also simplify the for 
mulations by replacing where possible a hypothetically 
formulated rule by one categorically formulated. Asa 
basis of this reformulation we take the rules of quality 
for Figs, i, 2 and 3, which have already been expressed 
categorically; viz. for Figs, i and 3: The minor pre 
miss must be affirmative, and for Fig. 2: The con 
clusion must be negative. Conjoining the categorical 
rule (of quality) for Fig. i with its hypothetical rule, 
If the minor is affirmative the major must be universal, 
we deduce for this figure the categorical rule (of quantity), 
The major must be universal. Again, conjoining the 



THE CATEGORICAL SYLLOGISM 



95 





c 

(U 


oT 


c 

0) 


o 






















: 




















> 






-t i 


















!H 


c > 

1 


rt 




| 








o 








fti 


55 JH 
r5 > 


^ 


g 


t/j 








< < 




- 




o 3 

II 

J^ 


8 c 


<*, o 


C/) 

s 




8 < 

l< 
s < 




S 

w 
< 




2 

W < 

w w 
< w 


HH O 

< S 







3 

w 
w 




cu 


3 

u 


In 


*rf 


p ^ 

< 


Q 

<^ 
w 




w < 



< w 
w < 


< w 




~ 

< < w 

*^ W < 




^ 


t: 




rbl 




^^ 










T 




CO Ctf 


CU 


in 


vl 














41 c r" 


g 


E ! 


I 

c/5 


|~ 

PH ^ 


O 

a, 
o 














oo oo oo 

* . e . IS . 


3 

So 


c 

S 


a 




cu 
















f T 


















x x 






Distribution for each I 


Fig. 2 
a. Conclusion negative. 


b. Major universal. 


r. If minor particular, then 
conclusion particular. 


of distribution to the s( 


en 

g 

O 

(U S 

> ss 

-S 1 

a; - 


tn" 
TJ 

8 

i"e 

O 

o v - 

s S 

2 

tn O 

C 
S 43 
43 

1 


^n" 

1 

CO 

p 


ll 

!l 

^ 4^ 
en O 
3 <-> 
S 43 

i? 


row (8 moods) 
i-numbered columns (12 moods 


two columns (6 moods) ... 


<U JJ <L> 
43 -5 43 

i ; i : i : 

i^!l: 

-d 

2! sl sl 


C/5 






c 
<u 


j 


^ en 

JS 


S 43 
4^ ^> 
tn en 


C/5 
C/5 


en en 


en en 


rt 


* e * S * S 

en 5 en S en p; 


CD 







c3 3 


u 


5 b 


CJ CJ 

cu cij 




OJ 


CJ CJ 
<U <U 


CJ 


CJ 


tl | U | U | 


rV 




* 




<L) 


C3 eu 


a? a? 


<D 


<5? 4? 


57 5? 





a? o 57 o oT" 


"rt 


:| 


05 
en 


|| 


C/) 


CT 1 


OJ CU 


CJ 


- j 


cu cu 





I-H M CO 

cu cu cu 


CJ 

QJ 

o, 

CO 
(U 

H 


f3 


S 


r. If minor p 
conclusiorj 


Applying 


O p 


3 p 


ci 




s s 




5" 



96 CHAPTER IV 

categorical rule (of quality) for Fig. 2 with its hypo 
thetical rule If the conclusion is negative the major 
must be universal, we deduce for this figure the cate 
gorical rule (of quantity), The major must be universal. 
Lastly, conjoining the categorical rule (of quality) for 
Fig. 3 with its hypothetical rule, If the minor is affir 
mative the conclusion must be particular, we deduce 
the categorical rule (of quantity) for this figure, The 
conclusion must be particular. The remaining rules 
must be repeated without modification. 

The Special Rules of Distribution for each Figure 
and the application of these rules of distribution to the 
scheme of possibly valid moods unrejected by the rules 
of quality are set out on the preceding page. 

14. We will now compare the results reached by 
the two methods direct and indirect. The direct 
method determines, by means of certain intuited dicta, 
what moods are to be accepted as valid ; the indirect 
method determines on equally intuitive principles 
what moods are to be rejected as invalid, and conse 
quently what moods remain unrejected. We gather from 
this comparison that the 24 moods (6 for each figure) 
that are established as valid by the direct method are 
identical with the 24 that are not rejected as invalid by 
the indirect method. It follows that the two methods 
must be used as supplementary to one another. For, 
apart from the use of the indirect method we should 
not have proved that the moods established as valid 
were the only valid moods ; and apart from the use of 
the direct method we should not have proved that the 
moods unrejected as invalid were themselves valid. In 
short, by the direct method we establish the conditions 



THE CATEGORICAL SYLLOGISM 



97 



that are sufficient to ensure validity, and by the indirect 
those that are necessary to ensure validity. 

1 5. The attached diagram, taking the place of the 
mnemonic verses, indicates which moods are valid, and 
which are common to different figures. The squares are 
so arranged that the rules for the first, second and third 
figures also show the compartments into which each 




mood is to be placed, according as its major, minor or 
conclusion is universal or particular, affirmative or nega 
tive. The valid moods of the fourth figure occupy the 
central horizontal line. 

1 6. A very simple extension of the syllogism and 
of the corresponding antilogism is treated in ordinary 
logic under the name Sorites, which is a form of argu 
ment comprised of propositions forming a closed chain ; 

J. L. II 7 



9 8 CHAPTER IV 

and may be defined as an argument containing any 
number of terms and an equal number of propositions, 
such that each term occurs twice and is linked in one 
proposition with one term, and in another with a different 
term. E.g. an argument of this form, containing five 
terms, would be represented by the five propositions: 
a b, b c, c d, d e, e a, where each term placed 
first may stand indifferently either for subject or for 
predicate. Now it will be found that the necessary and 
sufficient rules for inferences of this form are virtually 
the same as for the three-termed argument; viz. 

A. Rules of Quality. 

(a^) For an affirmative conclusion, all the pre 
misses must be affirmative. 

(# 2 ) For a negative conclusion, all but one of the 
premisses must be affirmative. 

B. Rules of Distribution. 

Any term recurring in the premisses must be 
distributed in (at least) one of its occurrences. 

Any term occurring in the conclusion must 
be undistributed, if it was undistributed in its 
premiss. 

The rules for the corresponding antilogism reduce to 
tw r o, viz. 

A. Rule of Quality : All but one of the propositions 
must be affirmative. 

B. Rule of Distribution: Every term must be dis 
tributed at least once in its two occurrences. 

17. There are certain irregular forms of syllogism 
or of sorites, which may be reduced to strict syllogistic 
form by the employment of certain logical principles, 
the nature of which we shall proceed to discuss. The 



THE CATEGORICAL SYLLOGISM 99 

arguments to be considered are those which involve a 
larger number of terms than of propositions; and it is 
necessary, in order to test the validity or invalidity of 
such arguments, to substitute if possible, for one or 
more of the propositions, an equivalent proposition, 
which will diminish the terms to the number of pro 
positions. This is done by means of obversion, con 
version, and other logical modes. Until this substitution 
is made, the argument may be valid, and yet break one 
or more of the rules of syllogism. Thus two of the 
premisses may be negative, and the argument yet be 
valid, the apparent violation of the rule being due to 
the presence of more than the proper number of terms ; 
for example, 

No right action is inexpedient, 
This is not a wrong action, 
/. This is expedient. 

Here by merely ob verting the two premisses we arrive 
at the standard syllogism of the first figure, namely : 

Every right action is expedient, 
This is a right action, 
.*. This is expedient. 

In all cases of substituting for a proposition some equi 
valent, we may require, besides simple conversion, the 
replacement of some term by one of its cognates. Thus 
in obversion, we replace P by not-/* or conversely ; P 
and not-P being the simplest case of cognates. Again 
any relative term may be replaced by its cognate co- 
relative. Now in the previous illustration obversion 
alone was required, whereas if the major had been 
written N o inexpedient action is right conversion would 
have been required before obverting. To illustrate the 

72 



ioo CHAPTER IV 

replacement of a relative by its corelative, we may take 
the old example from the Port Royal logic, 

The Persians worship the sun, 
The sun is a thing insensible, 
/. The Persians worship a thing insensible. 

This argument contains five terms, viz., the Persians, 
worshippers of the sun, the sun, a thing insensible, and 
worshippers of a thing insensible. The process of re 
ducing this argument to a strictly three-termed argument 
is effected by what is called relatively converting the 
major premiss, and again relatively converting the 
conclusion syllogistically arrived at from our new pre 
misses. The transformed argument then assumes the 
form of a strict syllogism in the third figure : 

The sun is worshipped by the Persians, 
The sun is a thing insensible, 
. .A thing insensible is worshipped by the Persians, 

where, by converting the conclusion, we reach that re 
quired, 

The Persians worship a thing insensible. 

1 8. The question whether the syllogism is actually 
used in thought process is met by noting that, while in 
ordinary discourse it is rare to find three propositions 
constituting a syllogism explicitly propounded, argu 
ments of a syllogistic nature are of frequent occurrence. 
These syllogisms are expressed as enthymemes, i.e with 
the omission of one at least of the requisite propositions. 
Now in an enthymeme there is one, and only one, pro 
position which could be introduced to render the corre 
sponding syllogism valid. For this reason the enthy 
meme is liable to one or other of two forms of attack : 
first it may be attacked on the ground that the premiss 



THE CATEGORICAL SYLLOGISM 101 

supplied by the hearer is true, and yet renders the argu 
ment invalid ; or secondly, that the premiss supplied by 
the hearer is false, and is yet the only one which would 
render the argument valid. The former case would be 
said to involve a formal fallacy; and the latter a material 
fallacy. For example: Consider the enthymeme, This 
flower is a labiate, because it is square-stalked. Here 
the premiss All labiates are square-stalked, which is 
true, renders the argument formally invalid ; on the 
other hand, the proposition which renders the argument 
formally valid, namely All square-stalked plants are 
labiates is false. These fallacies arise, for the most part, 
in the case of disagreement between disputants with 
respect to the conclusion. An enthymeme is free from 
both kinds of fallacy when the premiss to be supplied 
is known or accepted by all parties, and at the same 
time, renders the argument formally valid. Thus a 
strictly valid argument is expressed in the form of an 
enthymeme when there is no question with regard to 
the truth of the omitted proposition which will render 
the argument formally valid. 

This may be instructively illustrated by taking ex 
amples where either the major or the minor premiss in 
each of the first three figures of syllogism is omitted. 

Mr X is a profiteer, and therefore he ought to be 
super-taxed, 

this argument is acceptable on condition that the re 
quired major premiss All profiteers ought to be super- 
taxed is admitted. 

All bodies attract, therefore the earth attracts, 
this requires the minor premiss The earth is a body. 



102 CHAPTER IV 

Mr X ought not to be super-taxed, therefore he is 
not a profiteer, 

this requires the same major premiss as in the first 
example. 

Everyone present voted for Home Rule, therefore 
Mr Carson was not present. 

This requires as minor premiss Mr Carson would have 
voted against Home Rule. 

Mr Carson was present, therefore someone there 
must have voted against Home Rule. 

This requires the major premiss Mr Carson would vote 
against Home Rule. 

The earth is not self-luminous, therefore not all 
attracting bodies are self-luminous. 

This requires as minor premiss The earth is an attracting 
body. These three pairs of arguments are respectively 
in the first, second and third figure of syllogism. 

19. Having restricted my technical treatment of 
the syllogism to a single chapter, it will be easily in 
ferred that I attach considerable importance to this form 
of inference, while at the same time I hold it to be 
only one among many other equally important forms of 
demonstrative deduction. Syllogism is practically im 
portant because it represents the form in which persons 
unschooled in logical technique are continually arguing. 
It is theoretically important because it exhibits in their 
simplest guise the fundamental principles which underly 
all demonstration whether deductive or inductive. It is 
educationally important because the establishment of its 
valid moods and the systematisation and co-ordination 
of its rules afford an exercise of thought not inferior and 
in some respects superior to that afforded by elementary 
mathematics. 



CHAPTER V 

FUNCTIONAL EXTENSION OF THE SYLLOGISM 

i. THE categorical syllogism treated in the last 
chapter is correctly described as subsumptive. This 
term applies strictly to the first figure alone which 
may be called the direct subsumptive figure, and since, 
either by antilogism or by conversion, the other figures 
can be reduced to the first, these may be called indi 
rectly subsumptive figures. As explained in Chapter I, 
this form of inference employs in the most simple 
manner the Applicative followed by the Implicative 
principle. The ordinary subsumptive syllogism has a 
conclusion applying to the same range as the instantial 
minor, and its typical form is : 

Everything is/ if m ; 
This is m \ 
. . This is/. 

The first step in the extension of the ordinary syllogism 
to its functional form is to take a conjunction of dis 
connected syllogisms of the type : 
* Everything is/ if m ; This is m ; . . This is/. 
Everything is p if m f \ This is m f \ . . This is/ . 
Everything is/" if m"\ This is m n ; . *. This is./", etc. 
We next take m, m , m", etc., to be determinates 
under the determinable M, and /, / , /", etc., to be 
determinates under the determinable P. If then we 
can collect these major premisses into a general formula 
holding for every value of M and P in accordance with 



io 4 CHAPTER V 

the mathematical equation P=f(M), then we have an 
example of what may be called the functional exten 
sion of the syllogism, or (more shortly) of the functional 
syllogism, where the major or supreme premiss may 
be expressed in the simple form P=f(M}. Thus in 
the subsumptive syllogism the terms that occur in the 
minor and major premisses are merely repeated in the 
conclusion ; but, in the functional syllogism which yields 
an indefinite number of different conclusions for the 
different minors, the terms which occur in these dif 
ferent minors and conclusions are specific values of 
the determinables presented in the supreme premiss. 
Now it will be seen that no other principles are used 
in the functional syllogism, except the Applicative and 
Implicative, which together are sufficient to extend 
deduction beyond the scope of merely subsumptive 
syllogism. As a concrete example, let us take the 
formula of gravitation, which may be elliptically ex 
pressed Acceleration P varies inversely as the square 

of the distance MJ and written in the form />= . 
Then, by the Applicative Principle : 



and adding, as Minor Premiss : 

In this instance M= 7, 
we infer, by the Implicative Principle : 

1 In this instance P -^. 

Similarly when the value of M is n, the value of 
P will be y^Y, and so on. The same form of inference 
holds for two or more independent variables : thus 



FUNCTIONAL EXTENSION OF THE SYLLOGISM 105 

Boyle s Law may be written 7^=239 Pf 7 ; then, as 
before, we infer : 

When the value of P is 5, and the value of V is 2, 
the value of T will be 2390. 

When the value of P is 3, and the value of V is 7, 
the value of T will be 5019. 

2. A functional expression is of course familiar 
to the mathematician, but it will be important to ex 
amine the logical principles in accordance with which a 
universal functional formula operates in mathematical 
demonstration. In the first place we may observe that, 
as in ordinary syllogism, the supreme or major func 
tional universal must always have been ultimately 
established by means of inductive generalisation, and 
in the last resort from intuitive or experiential data. 
Further, the functional universal may be said to be a 
universal of the second order, because it not only 
universalises over every instance of a given value m, 
but applies also to every value of M. In deducing 
from the major l P=f(M) conjoined with the minor 
A certain given instance is m we reach the conclusion 
This given instance is p, where p is found from the 
equation ( p=/(m). Here it is to be observed: first, 
that this type of conclusion can be drawn, not only for 
the minor which predicates m, but also for minors which 
predicate any other value of M ; and secondly, that the 
character predicated in each conclusion is not merely 
what is predicated in the functional major, but a deter 
minate specification of this predicate. 

In the functional syllogisms that we shall consider 
in this chapter, the functional major is to be understood 
to express a factual rule, or more particularly a Law of 



106 CHAPTER V 

Nature. The general conception of a Law of Nature 
has been discussed (in Chapter XIV of Part I) under 
the head of the Principles of Connectional Determina 
tion. There it is shown that a typical uniformity or 
Law of Nature may be expressed in the form that the 
variations of a certain phenomenal character depend 
upon an enumerable set of other phenomenal characters ; 
of these the former is taken to be connectionally de 
pendent upon the others, which are connectionally 
independent of one another. A specific universal, which 
expresses such a relation of dependence may also be 
called a Law of Covariation ; for the nature of the 
dependence (say) of P upon ABC is such that all the 
possible variations of which P is capable are determined 
by the joint possible variations of A, B, C, which are 
themselves connectionally independent of one another. 
3. We have a special case of this relation of de 
pendence or covariation when the determined character 
can be represented as a mathematical function of the 
determining characters ; and it is this special case which 
gives rise to the functional syllogism. Now, in a func 
tional major expressed (say) in the form P=f(A, B, C\ 
it may in general be assumed that the correlation of 
these variables is such that, not only can the value of 
P be calculated from any assigned values of A, B, C\ 
but also, conversely, that the value of A can be cal 
culated from the values of P,B,C\ and that of B from 
the values of P, A, C ; and that of C from the values 
of P, A, B ; and similarly for a larger number of such 
connected variables. This process is expressed in mathe 
matical terms as solving the equation P=f(A, B, C), 
to find the value of A, which is thus calculated as a 



FUNCTIONAL EXTENSION OF THE SYLLOGISM 107 

certain function of P t B, C, and so on. A convenient 
symbolisation for these several equations will be as 
follows : 

P=f t (A,B,C): 

from which we calculate 
A =f a (P, B, C) ; B=f b (P, A, C) C=f e (P, A, B). 

It is here assumed not only that P (in mathematical 
phraseology) is a single- valued function of A 9 B, C y 
but also that in solving this equation to determine A 
or B or C respectively, these also are single-valued 
functions of the remaining variables. When this as 
sumption holds, we may speak in a special sense of the 
reversibility of cause and effect ; i.e. not only is the 
effect P uniquely determined by the conjunction of the 
cause-factors A, B, C; but also each of the cause- 
factors themselves, such as A, is uniquely determined 
by the effect-factor P in conjunction with the remaining 
cause-factors B and C. In the simplest cases reversi 
bility follows immediately from the form of the function 
as seen in the example given of Boyle s Law. Here 
we have a correlation between temperature, pressure, 
and volume, in which a constant, say k, is involved, 
and which assumes indifferently the form : 

pv kO kO 

e= k> OTV = J>t = ^ 

In this simple case, the multiplier k indicates the 
special form of the function which in the general case 
was represented by the unassigned but constant symbol 
f. An equation which, in this way, solves uniquely for 
all the variables is known as linear. But even in the 
case of non-linear equations we must be able to deter- 



io8 CHAPTER V 

mine, amongst the theoretically possible solutions for 
any one of the variables, that which is the sole factual 
value. In other words, a unique determination of all 
the variables, in terms of a given number of them, may 
be taken as expressing the actual concrete fact. 

4. In this connection it is important to note the 
number of variables entering into the functional formula. 
In Boyle s Law this number is three; i.e. there are 
three variables, any one of which is connectionally 
dependent upon the two remaining variables, so that 
the scope of dependence may be measured either as 
two or as three : for the functional formula contains 
three variables which are nationally independent of one 
another, namely p, v, 6 ; but of these two only are 
connectionally independent of one another. These two 
may be taken indifferently either as v and 6, or as / 
and 6, or as p and v, where, according to the alternative 
taken, p or v or is connectionally dependent upon 
the two others. In general, when there are r functional 
relations, connecting n notionally independent variables, 
then any n r of these can be taken as connectionally 
independent of one another, and each of the remain 
ing r as connectionally dependent jointly upon the 
others. 

Thus when n = S, and r = 3, the three functional 
relations may be symbolised : 



or adopting a shorter notation : 

p f p . abcde ; q =f q . abcde ; r f r . abcde. 

Such a trio of equations are taken to be implicationally 
independent of one another ; i.e. from neither one or 



FUNCTIONAL EXTENSION OF THE SYLLOGISM 109 

two of them could we infer the third. Otherwise the 
number three would reduce to two. Now the number 
of implicationally independent equations is necessarily 
the same as the number of connectionally dependent 
variables. Hence for the case under consideration we 
may express the three independent functional relations 
in either of four typical forms : 



! P=fv abcde\ 

2. af a . pbcde \ 
3- P=fp- abcqr i 
4. a=f a .pqrde; 


q =f q . abcde; 
q =f q .pbcde-, 
d=f d . abcqr; 
b=fa.pqrde-, 


r = f r . abcde. 
rf r . pbcde. 
e = f e . abcqr. 
c^fc.pqrde. 



The first trio expresses the three effect-factors sepa 
rately in terms of the five cause-factors jointly ; the 
second expresses one cause-factor and two effect- 
factors separately in terms of one effect-factor and four 
cause-factors jointly; the third expresses one effect- 
factor and two cause-factors separately in terms of 
three cause-factors and two effect-factors jointly ; and 
the fourth expresses three cause-factors separately 
in terms of three effect-factors and two cause-factors 
jointly. 

In illustration of this general principle we will con 
sider the Law of Gravitation, which may be formulated 



where A is the force of attraction of any two masses 
m^ and m^ whose distance is d, c being constant for all 
variations of m^ and m^ as well as of d. In any appli 
cation of the above formula we must first suppose m l 
and m^ to be constant, so that the variation of A de 
pends solely upon that of d. The algebraical equation 
here is, however, logically incomplete. In the first 



no CHAPTER V 

place, as regards the effect A, we must add the state 
ment that it is a force acting in the direction of the 
line joining m lt m*. In the second place, as regards 
the cause d, not only must the distance of the line 
joining m l and m z be taken as a cause-factor, but also 
its direction. 

In comparing the Law of Gravitation with Boyle s 
Law, the constants k, c, m^ m^ represent unchange 
able properties of the bodies concerned, while/, 6, v, d 
represent their changeable states or relations. It is 
necessary then to include amongst the independent 
cause-factors the permanent properties of bodies as 
well as their alterable states or relations. 

5. In our typical expression of a set of functional 
equations the number of variables taken to be con- 
nectionally independent was the same in all the several 
equations. But a very important type of connectional 
formulae is that in which equations enter involving 
different numbers of independent variables. Consider 
the following : 

Let a body be allowed to fall in vacuo. Here the 
two independent cause-factors are the mass (m) of the 
body and the distance (d) from which it falls to the 
earth. The effect-factors to be considered are the time 
(t) of falling, and the impulse (/) of the body upon the 
earth. Since out of the four variables m, d y p, /, two of 
them, namely m and d, are (as cause-factors) connec- 
tionally independent, the standard form in which both 
of these would enter into the function is 

/ f p . md and t f t . md. 
But, where the body falls in vacuo, the time (t] is in- 



FUNCTIONAL EXTENSION OF THE SYLLOGISM in 

dependent of the mass (m)\ Hence in this case the 
two formulae assume the form 

/ =ft - md and / f t . d. 

In this case, since the solution of the equations gives 
uniquely determined roots, we have : 



and (4) t=f t .d, (5) d=f d (t\ 

and, by substitution from (5) in (i), (2), (3) respectively, 

(6)p=f;.mt y ( 7 )m=f m .^ (%)t=f t .mp. 
Now, since, of the four variables m, d, p, t, any two 
except t and d may be taken as connectionally inde 
pendent, either one of the following pairs of connec- 
tional equations may be used, thus : 

Taking m and d as independents : p = f. md with / = f . d, 
wand/ : p=f.mt d = f.t, 

pan&d,, : m=f.pd t=f.d, 

/and/ \m = f.pt d = f.t, 

wand/ : d=f. mp t=f.mp. 

Giving to the unassigned functions their actual form, 
we have here 

(6) p = mgt and (5) d=\gt\ 

where the constant g stands for the acceleration 32 ft. 
per second. Solving these equations so as to express the 
effect- factors / and t in terms of the cause-factors m and 
d we have 




6. The example just given suggests a certain 
further characteristic of the connectional equations 

1 This illustrates the principle underlying the inductive method 
of agreement ; where m is eliminated as a cause-factor relative to the 
effect /, since a variation in m does not entail a variation in /. 



H2 CHAPTER V 

of applied mathematics. As will be seen in the above 
illustration, the connectional equations from which the 
deductive process derives other but equivalent equations 
are of a mixed nature as regards the variables that 
are taken as independents. Of the two equations 
p^mgt) and d=\gf, from which the other equations 
are derived, the former expresses the effect-factor / in 
terms of the cause-factor m and the other effect-factor t\ 
while the second expresses the cause-factor d in terms 
of the effect- factor t. It is therefore necessary to solve 
this pair of equations by an appropriate process in 
order to derive the pair of equations which express the 
effect -factors in terms of the cause-factors ; namely in 
the form 



What holds in this particular example may be gene 
ralised. Instead of separating variables that are given 
from those which have to be deduced, we have a set of 
equations (corresponding in number to the dependent 
variables) which all the variables taken together have 
to satisfy. Thus, in the above example, the equa 
tions were not at first expressed by taking a pair of 
cause-factors as independent upon which the pair of 
effect-factors depended, but the first of the two equa 
tions was taken from the pair in which m and / were 
supposed to be independent, and the other from the 
pair in which m and t, or p and / were taken as indepen 
dent. That the particular example of the falling body, 
originally taken to illustrate a different principle, should 
have lent itself to the principle now under considera 
tion, is more or less accidental, and we will now put 



FUNCTIONAL EXTENSION OF THE SYLLOGISM 113 

forward an example which more naturally exhibits this 
new speciality of a set of determining equations. 

Thus : consider the effect of mixing two substances 
at different temperatures in order to find the resultant 
temperature which will be reached when thermal equi 
librium has been established. Here we must take as 
the causally determining factors, the two initial tem 
peratures Jt 2 , and the two thermal capacities of the 
substances k^ and >. The factors to be determined are 
the heat //i entering into or passing from the one sub 
stance, and the heat ff z passing from or entering into 
the other, together with the final temperature 0. Now 
the equations that must be here used express the con 
ditions that are to be satisfied the effects not being, 
in the first instance, expressible as functions of the 
cause-factors. These equations of condition are the 
following three : 



from which we find 



rr_>-, rr_ 

~ 



4 

The solutions of these equations give the three values 
JFf 19 // 2 , and 9 (respectively) that were to be deter 
mined. Thus, in the final solution we have succeeded 
in expressing the factors to be determined in terms of 
the determining factors. But, in the equations express 
ing the conditions to be satisfied, the first two express 
an effect-factor as a function of two of the cause-factors 
and one of the effect-factors, and the third equation 

J. L. II 8 



n 4 CHAPTER V 

expresses one effect-factor as a function of another 
effect-factor. The use of equations of this kind is 
necessitated by the inadequacy of our knowledge of 
the precise temporal process by which the causal con 
ditions operate until the final issue is reached. Thus, in 
the actual process, heat will be passing to and fro from 
one to the other of the two substances, and this will 
entail a rise or fall of their temperatures in an in 
calculable way, which may be roughly expressed by 
suggesting that the quantum of heat entering the 
cooler body may be too great, so that the flow of heat 
will immediately be reversed; and this process might 
be conceived as involving even an infinite number of 
ingoings and outgoings of heat. What we know, how 
ever, is that at any stage of the process the heat that 
leaves one body must be equal to the heat that enters 
the other, whether this quantum is to be reversed in the 
next stage or not. It is this law which is expressed in 
our third equation, while the other two equations 
express a law or property, specific to the two substances, 
which correlates the effect upon the temperature with 
the quantum of heat which enters or leaves the body. 
What then we know, are these conditions of conserva 
tion of the total heat, and the several thermal capacities 
of the bodies, and from this knowledge the final effects 
can be calculated. It would appear, in fact, that the 
cases in which this logical principle is exhibited are 
those in which we know what is entailed in a final 
state of equilibrium, without having adequate know 
ledge for tracing in detail the perhaps oscillating pro 
cesses which take place in the lapse of time before the 
final state of equilibrium is reached. 



FUNCTIONAL EXTENSION OF THE SYLLOGISM 115 

We have illustrated this in the simplest case, where 
only two substances are mixed, but the reader will 
easily be able to construct the corresponding equations 
for any given number of different substances. In all 
cases the final or resultant temperature is equivalent to 
the arithmetic mean of the initial temperatures, each 
weighted by the corresponding thermal capacity. 
Thus 



Now having given an illustration from physics, we 
will give a closely analogous illustration from economics. 
The formula of covariation which connects the quantity 
of a commodity that is exchanged with its price is such 
that the two opposed parties shall be satisfied at the 
rate of exchange finally agreed upon. Now the formula 
of covariation on the side of demand is assumed to be 
connectionally independent of that on the side of supply. 
That which represents the economic attitude of the 
consumers depends solely upon their relative desires 
for different commodities, their monetary resources, 
and we may add the prices at which they are able 
to buy commodities other than that under consideration. 
In the same way, the attitude of the producers is wholly 
independent of that of the consumers; and depends 
upon the contract-prices current for the employment of 
the several agents of production, and upon the efficiency 
of these agents when co-operating in producing the 
commodity. It will thus be seen that the several 
factors that determine the conditions of supply are 
independent of those that determine the conditions of 
demand. Here, as in the case of thermal equilibrium, 

82 



n6 CHAPTER V 

the equations of condition express, not the effect-factors 
as functions of the cause-factors, but the conditions 
taken together which satisfy the consumers and the 
producers regarded each as economically independent 
of the other. 

This economic illustration differs from the case of 
thermal equilibrium in the important respect that the 
two functions of demand and supply respectively replace 
the actually operating cause-factors, which are highly 
complex and do not explicitly enter into the equations 
to be solved. 

7. The above illustrations of the functional ex 
tension of the syllogism have shown how, by the use 
of a set of functional premisses standing as majors, we 
may take not only minors which enable us to infer 
an effect-factor from the knowledge of a given cause- 
factor, but also minors which enable us to infer a cause- 
factor from the knowledge of a given effect-factor. The 
supposition upon which this is based has been called 
the Principle of Reversibility. We shall now show that 
it is this principle which underlies the so-called method 
of Residues, and other similar deductive processes. The 
canon of this method is stated by Mill as follows: 

Subduct from any phenomenon such part as is 
known by previous inductions to be the effect of cer 
tain antecedents, and the residue of the phenomenon is 
the effect of the remaining antecedents. 

In using the term subduct Mill intends no doubt to 
hint that, in the simplest cases, for subduct we may 
substitute subtract. Thus Jevons, in his Elementary 
Lessons, takes the case of ascertaining the exact weight 
of any commodity in a cart by weighing the cart and 



FUNCTIONAL EXTENSION OF THE SYLLOGISM 117 

load, and then subtracting the weight of the cart alone, 
which has been previously ascertained/ Here what 
corresponds to the effect (/) is the weight of the cart 
and load together, and its causes are (a) the weight of 
the commodity, and (b] the weight of the cart : so that 
the functional datum assumes its simplest form, viz. 
p = a + b, which by reversibility gives ap b. This is 
a case of solving an equation / =/(, b) to find a, and 
deducing af(b,p\ the equations being linear. The 
next simplest example of such reversibility is that of the 
composition of forces. Here the p 

diagonal OP represents the effect, 
and the sides OA, OB, the cause- 
factors. Just as, given the two 
cause-factors OA and OB, we 




parallel and equal to OB 
to find the effect OP ; so, given OA as one cause-factor 
and OP as effect, we may draw OB parallel and equal 
to AP to find the other cause-factor OB. Innumerable 
other examples may be given of reversibility for more 
or less complicated cases. But the classical example 
most frequently cited is the Adams- Le verier discovery 
of the planet Neptune from the observed movements 
of Uranus. Here we may represent the positions and 
masses of the Sun, of the Moon and of the unknown 
Neptune by the symbols #, b, c respectively; and the 
movement of Uranus by the symbol p. Thus p was 
theoretically known as a given function of a, b, c, say 
P ~f* ( a > ^ ^), where p stands elliptically for the effect, 
and a, b, c for the several cause-factors. The solution 
for c was then uniquely calculated in the form 



n8 CHAPTER V 

Now it will be observed that the so-called Method 
of Residues, which is based upon the assumption of 
reversibility is purely deductive, in that (i) it employs 
only the Applicative and Implicative principles of in 
ference, and (2) the conclusion obtained applies solely 
to the specific instances for which the calculation is 
made. This consideration shows that there is no justifi 
cation for putting Herschel s method of Residues under 
the head of methods of induction, along with such 
methods as those of Agreement and Difference; for, 
on the grounds above alleged, it is purely deductive. 
On this matter Mill sees half the truth ; for, in com 
paring the Method of Residues with that of Difference, 
he remarks that the negative instance in the former is 
not the direct result of observation, but has been arrived 
at by deduction. And again, in his formulation of the 
Canon of Residues, he speaks of such part of the 
phenomenon as is known by previous induction, where 
he fails to note that what is known by previous induction 
functions merely like the major premiss of a syllogism, 
and therefore does not in any way render the inference 
inductive. What holds for the method of Residues holds 
also of many less technical processes which, while 
purely deductive, have been obscurely conceived as 
inductive. For instance, the procedure in a judicial 
enquiry or by a police detective or of historical research 
in discovering the specific cause of a complicated set 
of circumstances constituting an observed effect, is 
purely deductive ; for it employs as major premiss 
known laws of human or physical nature under which the 
known circumstances are to be subsumed in the minor ; 
while the conclusion refers solely to the case sub judice. 



FUNCTIONAL EXTENSION OF THE SYLLOGISM 119 

8. Something should be said in explanation of the 
fact that inferences of this kind are so frequently spoken 
of as inductive. It is not only because the major premiss 
must itself have been obtained by induction, but further 
because the minor premiss represents a fact obtained 
by observation, that logicians have made this mistake ; 
for the notion of observation or experimentation as the 
method by which new knowledge is acquired is in 
variably associated with induction. But it should be 
pointed out that there is here a confusion between the 
matter and the form of an inference. Mere syllogism 
will obviously yield new material knowledge, provided 
that the minor premiss represents new material know 
ledge such as can only be obtained by observation. 
For example, from the observation that the importation 
of food has been taxed, we may infer the new material 
knowledge that the price of food will rise at a certain 
time in a certain economic market, if we have been 
otherwise assured of the major premiss appropriate to 
the circumstance. The form of such an inference is 
purely deductive, and the fact that historical research 
and not a merely foreknown universal formula has 
been required to establish the minor does not render 
the argument in any sense inductive ; for the conclusion 
holds only of the period and region to which the causal 
occurrence which has been discovered applies, and does 
not involve any inductive generalisation from one period 
or region to others. A further explanation of this com 
mon error is to be found in the fact that the conclusion 
reached deductively for a given instance may often 
be verified by awaiting the occasion for observing the 
effect in that instance. Now this process of verification 



120 CHAPTER V 

merely assures us that we have adequately estimated 
the causes operating in the given instance; but it has 
been almost invariably confused with the process of 
verifying, or rather confirming, the major premiss itself 
regarded as a problematic hypothesis as yet unproven. 
9. We ought now to distinguish, in these functional 
extensions of the syllogism, the element which is purely 
subsumptive from that which is functional ; for the two 
elements are practically always united in any concrete 
inference of the functional kind. It will be found that 
the factual formulae used in applied mathematics as 
major premisses for deduction necessarily involve two 
kinds of constituent, one of which is known as variable 
and the other as constant. The mathematical use of the 
term const ant presents certain difficulties from the logical 
point of view. There are certain constants e.g. the 
specific integers and the algebraical operators which 
are absolutely constant in the sense that in all their 
occurrences they stand for the same thing and are 
entirely independent of context. But those so-called 
constants which are dependent upon context are only 
referentially constant, being actually variable in precisely 
the same sense as the symbols that mathematicians 
recognise as variable. To explain this we may select 
illustrations from an innumerable variety of formulae 
used in applied mathematics. Consider, for instance, the 
formula which expresses the elasticity of a solid body 
which can support tension. The rule upon which the 
extension of such a body depends is shortly expressed 
in the formula T kE, where T stands for the variable 
tension and E for the variable extension; while k, 
which is said to be constant, stands for the elasticity of 



FUNCTIONAL EXTENSION OF THE SYLLOGISM 121 

the particular kind of solid for which the rule holds. 
Now this coefficient of elasticity, though constant for 
all possible variations of extension and tension of the 
body, yet varies from one kind of solid substance to 
another. We shall show, then, that such a coefficient, 
which mathematicians call constant, is used in the 
deductive process subsumptively, while that which is 
explicitly regarded as variable is used functionally. We 
may mark the real variability of a so-called constant by 
a subscript indicating the specific kind of substance of 
which the coefficient can be predicated. Thus k s will 
stand for the coefficient of elasticity of the kind of sub 
stance named s ; while k^ (say) will stand for that holding 
for the kind of substance called /. To express the 
mathematical procedure in strictly explicit logical form : 

Major Premiss. Every body, say b, which is k s has 
the property expressed by the algebraical equation 
T=k s E. 

Minor Premiss. A certain body b is k s . 
Conclusion. The body b has the property expressed 
by the equation T=k s E. 

Now this is a merely subsumptive syllogism, in which 
the coefficient k s and the body b recur unmodified in 
the conclusion as in the premisses. Thus, the coefficient 
which is called constant is used solely in a subsumptive 
form of syllogism ; but, inasmuch as a similar formula 
applies to bodies of a different nature (such as /), the 
coefficient k is not absolutely constant but varies 
according to the substance of the solid. In logical 
analysis, we must recognise the distinct ways in which 
the so-called constants and the so-called variables enter 
into the deductive process. This may be expressed 



122 CHAPTER V 

logically by defining the order in which the variations 
have to be made. For we haveyfr^ to consider varia 
tions of the so-called variables, which determine the 
range of the conclusion as holding for every case over 
which the constant applies. Only after this range of 
variation has been taken into consideration may we 
proceed to vary the so-called constants, and for any 
new value carry out the same range of variations of the 
variables. In language borrowed from mathematical 
terminology, we may say that the variations of the 
explicit variables are to be made within the bracket, 
while the variations to be made of the so-called constants 
are to be made outside the bracket. 



CHAPTER VI 

FUNCTIONAL DEDUCTION 

i. UNDER this heading we shall discuss the prin 
ciples underlying the deduction of formulae in the 
sciences of mathematics and logic. Although properly 
speaking pure mathematics is a development of logic, 
yet certain important points of distinction between the fyu 

two sciences must be brought out. It has been very 
commonly assumed that the sole method of deductive 
procedure in pure mathematics, including Geometry, 
is syllogistic. Now although it will be found that no 
fundamental principle is employed in mathematical de 
duction other than the Applicative which is essential 
for syllogism yet the conclusions successively derived 
from previously established formulae are not such as 
could be inferred by means of any mere chain of syl 
logisms. To explain this, it is necessary to point out the 
peculiar nature of the relation between conclusion and 
premisses in mathematical processes. Ordinary syl 
logism, as has been explained, is of the comparatively 
simple type denominated subsumptive. If subsumptive 
inferences only were used in algebra or geometry, it 
would be impossible to demonstrate conclusions except 
for special cases subsumable under the primary intuited 
axioms or under some previously established formulae. 
Thus from such premisses as: Everything that is m is 
/ and Everything that is n is q we could infer sub- 



I2 4 CHAPTER VI 

sumptively only that Every thing that is m and n is / 
and q! In other words, by means of subsumptive de 
duction, we can infer only that what holds universally 
of the members of a genus, m or n, holds universally of 
the members of their common species, viz. of the things 
that are characterised as being both m and n. For 
example : in geometry, having established a formula for 
all triangles and a formula for all right-angled figures, 
we could by merely subsumptive inference predicate of 
any species of triangles say right-angled only what 
could be predicated of all triangles ; and similarly we 
could predicate of any species of right-angled figures 
say three-sided only what could be predicated of all 
right-angled figures. But actually in geometry we prove 
a property (viz. the Pythagorean) of all right-angled 
triangles which is not the same as any universal pro 
perty either of three-sided or of right-angled figures. 
Similarly in algebra, we can deduce properties of all 
integers divisible by 2 and divisible by 3, which hold 
neither of all integers divisible by 2 nor of all integers 
divisible by 3. A predicate which holds for all members 
of a species, but not for all members of any genus to 
which by definition the species belongs, is technically 
known as a proprium or tStoz/, either of which term may 
be translated property. It is one of the special objects 
of this chapter to analyse the process by which proper 
ties, in this technical sense, are deduced. It will be 
shown that, in the deductions peculiar to pure mathe 
matics, the premisses and conclusions assume the form 
of functional equations; and that it is owing to this 
characteristic that properties in the technical sense can 
be deductively demonstrated. We therefore give the 



FUNCTIONAL DEDUCTION 125 

name functional deduction, in antithesis to subsumptive 
or syllogistic deduction, to the specifically mathematical 
form of inference. 

2. Before entering upon the main discussion it 
will be well further to consider the nature of the Aris- 
totelean tSioz>. Many modern logicians have failed to 
grasp the important significance to be attached to this 
notion. Elementary textbooks, such as that of Jevons, 
define a property of a class as any character not in 
cluded in the connotation, which can be predicated of 
all, as distinct from an accident which can be predicated 
only of some, members of the class. On the other hand, 
Mill attempts to define a proprium in closer connection 
with the scholastic development of Aristotle s doctrine, 
and distinguishes not merely between an invariable and 
a variable predicate of a class which satisfies Jevons 
but defines a proprium as a predicate not included in 
the connotation of the class (and therefore assertible in 
a proposition not merely verbal) but following neces 
sarily from the connotation alone. But since a pro 
position which merely asserts connotation is verbal, 
this account of the proprium is incompatible with the 
theory so clearly expounded in his chapters on Defi 
nition and on Verbal Propositions that no conclusion 
can be drawn from merely verbal propositions that is 
not itself merely verbal. From this it follows that in 
order demonstratively to establish any invariable charac 
ter that can be regarded as necessary, we require as 
premisses not only definitions but also real or genuine 
propositions, and, in mathematics, ultimately axioms. 
It is true that Mill distinguishes two ways in which the 
proprium may follow necessarily from the connotation: 



126 CHAPTER VI 

it may follow as a conclusion follows premisses, or it 
may follow as an effect follows a cause. But this dis 
tinction is purely illusory and wholly irrelevant to the 
notion of necessity of demonstration ; for, in both cases, 
the ground for Mill s account of a proprium as neces 
sarily following from the connotation is that appropriate 
knowledge will enable us to infer demonstratively the 
proprium from the connotation. A legitimate distinc 
tion may be drawn according as the major premiss from 
which a proprium is inferred is of the nature of an axiom 
or of a causal law. Indeed Mill himself goes on to say 
that the necessity attributed to the proprium means that 
its not following would be inconsistent with i.e. its 
following could be inferred from either an Axiom or 
a Law of Nature. Thus in both cases the notion of 
following \s the same, and simply means inferrible from. 
The proprium, therefore, never follows from the conno 
tation alone, but requires in addition one or other of 
the two species of real propositions, axiomatic or ex 
periential, to serve as major premiss. 

3. The functional equations used in the deductions 
of pure mathematics in some respects differ from and 
in others agree with those used as major premisses in 
the process discussed under the head of the functional 
extension of the syllogism. The equation used in this 
latter process serves as a single major premiss for a 
number of specific conclusions found by replacing the 
variables by their specific values. Here the functional 
equation assumes the form P=f(A, B, C) for all values 
oiA,B, C. But the equations used in the process of func 
tional deduction are of the form/(^, ,C) = (f>(A, B, C) 
for all values of A, B, C, where all the variables are 



FUNCTIONAL DEDUCTION 127 

independently variable, and the equation therefore 
contains no such symbol as P that can be exhibited as 
dependent upon the others. The distinction between 
these two types of equation is familiar to mathematicians ; 
the former may be called a limiting, the latter a non- 
limiting equation. The limiting equation is generally 
used to determine one or other of the quantities P, A, 
B, or C, in terms of the remainder; so that here we 
associate the antithesis between dependent and inde 
pendent with the antithesis between unknown and 
known; whereas, in the non-limiting equation, no one 
of the variables can be regarded as unknown and as 
such expressible in terms of the others regarded as 
known. The distinctions that have been put forward 
between these two types of functional process are tanta 
mount to defining the functional syllogism as that which 
proves factual conclusions from factual premisses, and 
functional deduction as that which proves formal conclu 
sions or formulae from formal premisses, i.e. from 
formulae previously established. It will further be ob 
served, from the simple illustrations which follow, that 
whereas the functional syllogism requires only the one 
functional equation that serves as major premiss, the 
process of functional deduction will necessarily involve 
a conjunction of two or more functional equations, all 
of which are, as above explained, formal and not 
factual. 

To illustrate the general formula used in functional 
deduction, viz. : 

/(, b y c, ...) = <(#, 6, c, ...) 
which is understood to hold for every value of the 



I2 8 CHAPTER VI 

variables A, B> C, ..., we may instance the following 
elementary examples : 



and axb bxa, 

both of which involve two variables; and again 



and (a + 6)xc = (axc) + (dx c), 

both of which involve three variables. The last three 
formulae are known respectively as the Commutative, 
the Associative and the Distributive Law. 

4. In the functional equations of mathematics it is 
important to realise the range of universality covered by 
any functional formula. This range depends upon the 
number of independent variables involved in the formula, 
the range being wider or narrower according as the 
number of independent variables is larger or smaller. 
For example, supposing that JT, y, z have respectively 
7, 5, i o possible values; then the number of applications 
of the formula involving ^ alone is 7, that of a formula 
involving ^ and y alone is 35, and that of a formula 
involving .r and y and z is 350. And in general, the 
number of applications of a formula is equal to the 
arithmetical product of the numbers of possible values 
for the variables involved. Now the number of possible 
values of any variable occurring in logical or mathe 
matical formulae is infinite; hence, for the cases re 
spectively of i, 2, 3... variables, the corresponding 
ranges of application would be oo , oo 2 , oo 3 ..., consti 
tuting a series of continually higher orders of infinity ; 
or rather, in accordance with Cantor s arithmetic, each 
of the ranges of application for i, 2, 3 ... variables is a 



FUNCTIONAL DEDUCTION 



129 



proper part of that for its successor, although their 
cardinal numbers are the same. 

Now it will be found that, in inferences of the nature 
of functional deduction, the derived formula may have 
a range of application not narrower than but equal 
to or even wider than that from which it is derived. 
Thus the word deduction as here applied does not 
answer to the usual definition of deduction (illustrated 
especially in the syllogism) as inference from the generic , 
to the specific ; although the only fundamental principle 
employed in the process is the Applicative, according 
to which we replace either a variable symbol by one of 
its determinates or one determinate variant by another. 
But here a distinction must be made according as the 
substituted symbol is simple or compound. If we merely 
replace any one of the simple symbols a, b, c by some 
other simple symbol we shall not obtain a really new 
formula, since the formula is to be interpreted as holding 
for all substitutable values, and hence it is a matter of 
indifference whether we express the formula in terms of 
the symbols a, b, c, (say) or of/, q,r. In order to deduce 
new formulae, it is necessary to replace two or more simple 
symbols by connected compounds. 

For those unfamiliar with mathematical methods, it 
should be pointed out that, when any compound symbol 
is substituted for a simple, the compound must be en 
closed in a bracket or be shown by some device to 
constitute a single symbolic unit. Though we may 
always replace in a general formula a simple by a com 
pound symbol, the reverse does not by any means hold 
without exception. The cases in which such substitu 
tion is permissible have been partially explained in the 

J. L. II 9 



I 3 o CHAPTER VI 

chapter on Symbolism and Functions. There it was 
shown that, if a formula involves such compound 
symbols or sub-constructs as f(a, <), f(c, d) etc., and 
only such, where none of the simple symbols used in 
the one bracketed sub-construct occur in any of the 
others, then these bracketed functions are called dis 
connected. It is in the case of disconnected functions 
that free substitutions of simple symbols for the com 
pound are permissible. The reason for this is that, for 
the notion of a function of any given variants, it is 
essential that these shall be variable independently of 
one another. Now, when the different sub-constructs 
or bracketed functions are connected with one another 
through identity of some simple symbol, say a, it is 
clear that we cannot contemplate a variation of one of 
these compounds without its involving a variation of the 
other connected compounds. Hence we should be vio 
lating the fundamental principle of independent varia 
bility of the variants, if we freely substituted for such 
connected compounds simple symbols which would have 
to be understood as capable of independent variation. 
Hence, it is only when the various compounds involved 
in a function are unconnected, that for each of such 
compounds a simple symbol may be substituted. 

5. Returning to the problem under immediate con 
sideration, a simple illustration from algebra will show 
how, by making appropriate substitutions in a given 
functional formula, we may demonstrate a new formula. 
Thus, having established the formula that for all values 
of x and y 

(i) (x+y)*(x-y)=x>-f 

we may substitute for AT and y, respectively, the connected 



FUNCTIONAL DEDUCTION 131 

compounds a + b and a b ; and so deduce (by means of 
the distributive law for multiplication etc.) that for all 
values of a and b, 

(ii) wd = (a + b) 2 -(a-b)\ 

This is a new formula, different from the previous one, 
because the relation between a and b predicated in 
(ii) is different from the relation between x and y pre 
dicated in (i). Moreover the range of application for 
(ii) is no narrower than that for (i); for (i) applies for 
every diad or couple l x toy, and (ii) for every diad or 
couple a to < ; and therefore the ranges for (i) and (ii) 
are the same. Again, ,if we have established the Com 
mutative, Associative, and Distributive formulae given 
above, the reader will see that, by means only of the 
Applicative principle, we can deduce from these three 
formulae what is in fact a new formula: 

(a + 6)(c + d)=*ac+6c + ad+ bd. 

In this case, the formula deduced has a wider range of 
application than any of the formulae from which it is 
deduced. For the premisses for this deduction involve 
respectively 2, 3 and 3, independent variables, while 
the conclusion involves 4; showing, as explained in the 
previous paragraph, that the range of application of the 
conclusion is wider than that of even the widest premiss. 
To reach a conclusion inclusive of and wider than the 
premisses is in general considered the mark of an in 
ductive inference; but we have shown by the above 
example that, where the premisses are functional formulae 
involving more than one independent variable, the mere 
employment of the Applicative principle enables us to 
reach a formula wider than any of the premisses. Now 

92 



I32 CHAPTER VI 

it is in accordance with general usage to define deductive 
inference as that which employs no principles but the 
Applicative and the Implicative. In the purely deduc 
tive process of mathematics, in fact, it is only the Appli 
cative principle that is required ; and pure mathematics 
is regarded as specially typifying the power of mere 
deduction. It is true, however, that mathematicians 
have employed a method which involves also the Impli 
cative principle, viz. what has always been known under 
the name of mathematical induction. In these later 
days, this method has been regarded as more specifically 
characteristic of mathematics than any other. But the 
line of distinction between induction and deduction, in 
their extended potentialities fordemonstrative inference, 
cannot be drawn on any logical principle that would be 
universally accepted. It is for this reason that I have 
attempted to treat in one large division of my Logic all 
varieties of demonstrative inference, on the ground that 
it is the demonstrative character of these inferences that 
brings them within one sphere, and that the distinction 
that might be drawn between deductive and inductive 
demonstration has no important logical significance com 
parable with that between demonstrative and proble 
matic inference. Mathematics, as the above adduced 
inferences illustrate, provides a host of cases in which 
the Applicative principle alone is explicitly employed 
without any recourse to the Implicative principle. These 
inferences might be called purely Applicative 1 in con 
trast to the syllogism, which in our analysis has been 
shown to involve the Implicative as well as the Appli 
cative principle. Again the construction of the logical 
1 Cf. Chapter I, p. 1 1 and onwards. 



FUNCTIONAL DEDUCTION 133 

calculus involves the Implicative as well as the Appli 
cative principle, and will be discussed later. Before 
proceeding to this topic, we must complete our account 
of mathematical demonstration by an analysis of mathe 
matical induction, which also involves both principles. 
6. Mathematical induction assumes a unique place 
in logical theory. It resembles other forms of demon 
strative induction, which will be discussed in a later 
chapter, where it will be shown that the universal mark 
of this type of induction is that the conclusion demon 
stratively inferred asserts for every case what has been 
asserted in one premiss for a single case. The possi 
bility of such demonstration rests upon the logical 
character of the other premiss, which may be of different 
types, each type yielding a different form of demonstra 
tion. The distinctive characteristic of mathematical in 
duction is that it is concerned with finite integers. These 
constitute a discrete series beginning with the integer i, 
and proceeding step by step in the construction of suc 
cessive integers. The generation of each successive 
integer from the preceding is indicated by the operation 
plus i. Thus, using the illustrative symbol n to stand 
for any finite integer, the operation symbolised as n -f i 
will yield the next following integer. This construction 
defines the general conception of a finite integer which 
is fundamental for arithmetic. The method of mathe 
matical induction introduces the notion of function. 
Thus f(n) will be used to stand for any proposition 1 

1 The functions previously adduced were mathematical, i.e. con 
structs yielding quantities, whereas the function here introduced is pro- 
positional, i.e. a construct yielding a proposition. And, in general, the 
equating of two mathematical functions yields a prepositional function. 



I 34 CHAPTER VI 

about the specific integer n, where variation of form 
will be represented by changing/ into < say, and varia 
tion of reference by changing n into m say. The argu 
ment in its general form will consist of the following 
assertions of two premisses and of the inferred con 
clusion : 

Implicative Premiss: The proposition f(n) would 
imply the proposition /(#+i) for every finite in 
teger n. 

Categorical Premiss : ( /(i) holds. 

Conclusion: Therefore c f(n) holds for every finite 
integer n. 

In this argument we observe that the conclusion states 
categorically what is stated hypothetically in the im- 
plicative premiss; and further that it predicates for 
every case what is predicated for a single case in the 
categorical premiss. Its demonstrative force may be 
shown by resolving the argument into a succession of 
steps. Thus, by the applicative principle, we may re 
place in the implicative premiss n by i, and this yields 
the assertion *f(i) would imply/(2) ; then, adding the 
categorical premiss /(i), we infer, by the implicative 
principle, /"( 2 )- Again, replacing n by 2, /( 2 ) would 
imply /($)? and, adding to this the conclusion of the 
preceding inference, we may infer /($). If this process 
is indefinitely continued we are enabled, by use merely 
of the applicative and implicative principles, to infer 
successively /(2),/(3),/(4), etc., for every finite integer. 
The whole argument therefore rests merely upon the 
same principles as are involved in ordinary deduction; 
and yet the inference is of the nature of induction, 
because the conclusion is a generalisation of the same 



FUNCTIONAL DEDUCTION 135 

formula that the categorical premiss lays down only for 
a single case. 

The following is a simple application of mathematical 
induction : 

Let f(n) stand for the proposition: The sum of 
the first n odd integers = ;zV We have first to establish 
the implicative premiss, viz., 

l f(n) would imply f(n+ i). 
Nowf(n) is the proposition 

i+3 + 5 + 7 + -+(2-i) = *Y 
and/"(/z-f i) is the proposition 



Here the left hand side of the equation f(n -f 1) is ob 
tained from that off(n) by adding (2/2+ i). 

Hence, by the formula for the square of the sum of 
two numbers: viz., 



the conclusion is established that 

c \if(n) holds, then/(^+ i) would hold. 

Now/(i) holds; for i = i 2 . (Also 7(2) holds; for 
i+3 = 2 2 : and /(a) holds; for 1+3 + 5 = 3 2 .) 

Hence, having established the implicative premiss 
/(*) would imply/ (^+ i), and the single categorical 
premiss /(i), the required universal /(#) has been 
proved. 

7. In this account of the principles employed in 
establishing general algebraical formulae, special em 
phasis has been laid on the novelty of the conclusion 
as compared with the familiarity and obviousness of the 






136 CHAPTER VI 

premisses (including the axioms) from which the con 
clusion is drawn. This summary account of the methods 
and results of deductive reasoning enables us to meet 
what has been called the paradox of inference in a more 
direct way than that explained in Chapter I. For the 
existence of the mathematical calculus, where the con 
clusions are absolutely unknown to those who start by 
admitting as self-evident the fundamental premisses, 
constitutes a direct refutation of the arbitrary dictum 
that for valid inference the conclusion must not contain 
more than what is already known in asserting the pre 
misses. 

The notion of a calculus is generally associated with 
elaborate symbolism, which renders possible the more 
complex deductive processes in logic and mathematics. 
As a question of history, there is no doubt that the in 
troduction of such simple symbols as + , , x , created 
a revolution in mathematical science, and rendered it 
possible to make advances otherwise unattainable. 
Again it is an equally noteworthy historical fact that 
the best formal logicians, such as Leibniz and Lambert, 
were comparatively unsuccessful in their attempt to 
develop a logical calculus, which was first started by 
Boole on lines followed by all subsequent symbolists 
who advanced the science. Boole s method was simply 
to import the familiar symbols of elementary arithmetic 
into logic, making use of the fundamental formulae with 
which algebraists were already conversant. In this way 
he created the first great revolution in the study of 
formal logic, and one that is comparable in importance 
with that of the algebraical symbolists in the sixteenth 
century. I think, however, that Boole s procedure has 



FUNCTIONAL DEDUCTION 137 

led to considerable confusion with regard to the relations 
between the logical and the algebraical calculus, inas 
much as he seems to have supposed in common with 
many logicians of his time that the advance achieved 
by introducing mathematical formulae into logic made 
logic into a department of mathematics. This attitude 
of Boole s obstructed, for a considerable period, the in 
vestigation of the foundations of mathematics, which 
demanded the reversal of the relationship between the 
two sciences. It is under the influence mainly of Peano 
and of the new mathematicians such as Cantor, that we 
now recognise mathematics to be a department of logic. 
The current phrase mathematical logic is ambiguous 
inasmuch as it may be understood to mean either the 
logic of mathematics or the mathematics of logic. Now, 
in my view, the logic or rather philosophy of mathe 
matics is a study which ought to dispense entirely with 
symbolic language. It must, of course, explain the nature 
of symbols and of symbolic methods, and account for 
the extraordinary power of symbolism in deducing with 
absolute security previously unknown formulae. But 
the philosophical exposition of the deductive power of 
mathematics must be treated in language the under 
standing of which requires thought of a profounder 
nature than that required in merely following symbolic 
rules. As indicated in the chapter on Symbolism and 
Functions, the essential purpose of symbolism is to 
economise the exercise of thought; and thus symbolic 
methods are worse than useless in studying the philosophy 
of symbolism or of mathematics in particular. The phrase 
mathematics of logic, on the other hand, merely in 
dicates a certain line of development of logic, in which 



I 3 8 CHAPTER VI 

deductive processes are reduced to strictly demonstrative 
form by means of a symbolism founded on explicitly 
logical axioms. The important advances in this direction 
have been systematised with extraordinary success in 
Whitehead and Russell s great work Principia Mathe- 
matica, where it is shown how pure mathematics can be 
actually developed from pure logic. The value of the 
work consists, therefore, in reducing mathematics to 
logic, and not at all in reducing logic to mathematics. 
I shall attempt hardly any criticism of their formal de 
velopment of the science, and shall here confine myself 
to the principles which enter into its very elementary 
foundations. 

8. In contrasting the mathematical developments 
of logic with the ultimate foundations of the science, it 
will be convenient to use the terms premathematical 
and mathematical logic, the latter of which introduces 
certain novel conceptions, strictly formal in character, 
in addition to those employed in the former. There 
are certain notions common to the premathematical and 
mathematical departments of logic, and of these we have 
already discussed the nature of functions, illustrative 
and short-hand symbols, variables, brackets, etc., which 
before Peano and Russell had not received adequate 
recognition in logical teaching; they apply, however, 
over a wider field than mere mathematics, and must 
therefore be transferred without modification from the 
narrower science back to logic. The term formal as 
applied to these conceptions means that they are to be 
understood by the logician as such, and they include, 
besides those primitive ideas which are to be understood 
without definition, also derivative ideas which are com- 



FUNCTIONAL DEDUCTION 139 

pletely defined in terms of primitive ideas. For example, 
the notions of implication, alternation, disjunction and 
negation are formal, and of these we may take negation 
and alternation as understood without definition, while 
the others can be defined in terms of these two 1 . Again 
logical categories and sub-categories such as substantives 
proper, primary and secondary adjectives and proposi 
tions, come under the head of formal conceptions. There 
are also specific adjectives and relations, such as true, 
probable, characterised by, comprised in, identical 
with, which are formal; and, though some of them are 
ultimately indefinable, the understanding of all of them 
is essential to logical analysis. In contrast to these 
formal adjectives, such adjectives as red, hard, popular, 
virtuous, etc. are termed material, because their meaning 
is unessential to the explication of logical forms. In 
premathematical logic formulae are established for all 
adjectives as such, or for a limited set of adjectives 
comprised in such a sub-category as that of secondary 
adjective. The range over which these formulae hold 
must be said to be material, though it necessarily com 
prises adjectives which may happen to be formal, i.e. to 
have specifically logical significance. Passing from pre 
mathematical to mathematical logic, we find that new 
specific adjectives, having essentially logical significance 
and coming therefore under the head of formal concep 
tions, are introduced. We may specially mention in 
tegers and ratios. Integer is a logical sub-category 

1 This, at any rate, is the procedure of the Principia Mathematical 
but, while undoubtedly permissible from the point of view of the 
logical calculus, it is open to serious philosophical criticism, which I 
have given elsewhere. 



I 4 o CHAPTER VI 

comprised in the general category adjective ; and ratio is a 
logical sub-category comprised in the general category 
relation ; but what constitutes the new feature in mathe 
matical logic is that each specific integer and each 
specific ratio has itself essentially logical significance, 
while at the same time formulae hold for all integers 
and again for all ratios. Premathematical logic on the 
other hand can only establish formulae holding for ad 
jectives in general or for secondary adjectives in general. 
This distinction carries with it the further result that 
premathematical logic can only use illustrative adjectival 
symbols as variables over a. range of variation covering 
the whole category adjective, or the whole sub-category 
secondary adjective , while in mathematical logic there 
occur illustrative symbols for variables covering the 
range, in the one case integer, in the other ratio. Con 
sider for example such an illustrative symbol as m in 
ordinary or premathematical logic. The specific values 
that can be substituted for this variable are material; 
for the formal character of such of them as have speci 
fically logical significance is irrelevant to the truth of 
the formulae. In mathematical logic, on the other hand, 
all the specific values which can be substituted for a 
symbol m standing for any integer, say, or a symbol / 
standing for any ratio, denote formal conceptions. Again 
it is obvious that, besides the formulae which hold for 
adjectives in general, there are innumerable additional 
formulae holding for integers or for ratios; and this 
accounts for the variety and complexity of mathematics 
as compared with premathematical logic. But the es 
sential distinction between the two sciences or rather 
the two departments of logical science lies in the point 



FUNCTIONAL DEDUCTION 141 

already urged, namely that every specific adjective within 
a certain mathematical range has itself a logically de 
termined value ; whereas no logically determined value 
can be assigned to adjectives in general which enter 
into premathematical logic. This distinction may be 
summed up in other words by taking the two antitheses 
material and formal, and constant and variable, which 
combined give the four cases formal variables, formal 
constants, material variables and material constants. 
Now premathematical logic uses formal constants and 
material variables (and also in Mr Russell s work material 
constants), but nowhere formal variables. On the other 
hand mathematics uses formal constants, material vari 
ables, and also formal variables. It is therefore the use 
of formal variables that fundamentally distinguishes 
mathematics from premathematical logic. 

9. To continue our account of the relation between 
the premathematical and mathematical departments of 
logic, we must next define and illustrate the nature of 
those formal elements which are never expressed by 
variable symbols, and therefore come under the head of 
formal constants. To these, the name connectives will 
be given. The first division under this head includes 
what are known as operators in mathematics, such as 
plus, minus, multiplied by, divided by, as well as ana 
logous logical operators such as and, or, not, if. Thus 
the operation m + n, where m, n stand for determinate 
numbers, yields a certain determinate number; and 
analogously the operation p and qj where /, q stand 
for determinate adjectives, yields a certain determinate 
adjective. This is most clearly seen when a proper 
name has been invented to stand for the compound 



CHAPTER VI 

construct as well as for each of the constituents them 
selves: thus, the operation three-plus-five yields the 
number eight ; the operation rational-and-animated 
yields the adjective human. The analogy goes one 
step further when, in place of the simple predication 
yields, we use the complex yields-what-is-yielded-by : 
thus, the operation m plus n yields-what-is-yielded-by 
the operation n plus m ; the operation p and q yields- 
what-is-yielded-by the operation q and/. Now neither 
in logic nor in mathematics is it ever required to use 
illustrative or variable symbols to stand for formal 
operators like plus or and the reason being that no 
formula which holds for one operator will hold if we 
substitute indiscriminately any other operator. Hence, 
if symbols are used for formal operators, these come 
under the head of short-hand symbols, and never under 
the head of illustrative or variable symbols. Thus the 
operators both of logic and of mathematics enter as 
formal constants, never as variables. 

In the second division of connectives are to be in 
cluded certain relational predications which must be 
systematically illustrated and classified according to 
their different properties. Of these, the five of most 
fundamental importance are the relational predications : 
identical with, implied by, characterised by, comprised 
in, included in, together with their cognates. These 
are formal, and to represent them I shall introduce the 
short-hand symbols: t, t; X, X; ^, ; K, ; v, v respec 
tively. These five formal connectives are absolutely 
distinct from one another, although they have been 
frequently confused by logicians ; and this distinctive- 
ness is sufficient to account for the fact that they are 



FUNCTIONAL DEDUCTION 143 

never represented by variable symbols for which one 
could replace another. Thus: in the predication xly, 
the symbols x and y must stand for entities belonging 
to one and the same assigned category; but, in the 
predication xfy, x and y must stand respectively for an 
item or member and an enumeration or class; and, in 
the predication xy^y, x and y must stand respectively 
for a substantive and an adjective. Again, while I con 
nects entities belonging to any the same category, X 
connects only propositions or adjectives or relations; 
and v connects only classes or enumerations of the same 
order. And yet again : the relation identity is reflexive, 
symmetrical, and transitive; but the relations charac 
terised by ax\& comprised in are a-reflexive, a-symmetrical, 
a-transitive; while the relations implying and included 
in are reflexive and transitive but neither symmetrical 
nor a-symmetrical. The five connectives above enu 
merated may be said to be on the borderland between 
premathematical and mathematical logic. There are, 
however, many formal connectives which belong ex 
clusively to mathematics, of which the most funda 
mental is equals universally represented by the short 
hand symbol = . There is serious danger of confusing 
equal-to with identical-wit h because they agree in pos 
sessing the properties reflexive, symmetrical and tran 
sitive (to the consideration of which we shall have to 
return later). Other important connectives in logic and 
algebra are derivative from those above enumerated as 
fundamental. Classifying fundamentals and derivatives 
according to their properties we have the following 
table, where the initials F, S, T stand respectively for 
reflexive, symmetrical and transitive, and the suffix a 



144 



CHAPTER VI 



means for all cases, e for no cases/ and oi for some 
but not all cases. 



Formal Relations 

Identical, equal, co-implicant, coincident 

Differing by unity, co-opponent, co-remainder 

Greater, less, sub- and super-implicant, sub- and super- 
incident 

Sub- and super-opponent, sub- and super-remainder... 

Not greater, not less, implying, implied by, included in, 
including 

Other than, unequal, disjunct, alternant, co-exclusive, 
co-exhaustive 

Not disjunct, not alternant, not co-exclusive, not co- 
exhaustive 

Not implying, not implied by, not included in, not 
including 

Comprising, comprised in, characterising, charac 
terised by 



Properties 
a. S a T a 

* S a T e 
e S e 7 a 



F e S a T oi 

F a S a T^ 

F e S oi T oi 

F e S e T e . 



10. Not only formal constants but also material 
variables enter in the same way into mathematics as into 
premathematical logic. The particular values which any 
material variable may assume are unessential for pure 
logic and pure mathematics; and enter as significant 
factors only into applied logic or applied mathematics. 
For example, a variable representing any substantive 
or any adjective is replaced by a particular substantive 
or a particular adjective only when the general formulae 
established by logic are applied to concrete propositions. 
Similarly the purely formal notion of magnitude or of 
quantity, which enters into mathematics, is applied to 
several different species and sub-species such as mass, 
volume, intensities of different kinds, etc., the dif 
ferentiae of which, not being expressible in terms of 
pure mathematical conceptions, must be determined 
materially. Thus, for instance, in the mathematical 



FUNCTIONAL DEDUCTION 145 

formula 3^ + 5^ = 8^, q enters as a material variable 
standing for any quantity; and 3, 5, 8, = , + , as also the 
category q^t,antity itself, enter as formal constants. But in 
applying the material variable q to deduce the equation 

3 feet -f 5 feet = 8 feet, or 3 ohms -f 5 ohms = 8 ohms, 

the terms foot, ohm, as species of the genus quantity, 
have to be defined by means of conceptions outside 
the range of pure mathematics. In this way we see 
that variable symbols material as regards their range 
of application entering into premathematical and 
mathematical logic, assume their particular values when 
logical theorems are applied to experimental matter. 
Having shown then, as regards both formal constants 
and material variables, that general logic agrees in all 
respects with mathematics, the conclusion follows that 
the latter fundamentally differs from the former in the 
sole fact that it introduces formal variables. 

ii. Before examining the characteristics of the 
specifically mathematical notion * equals upon which 
its symmetry and transitiveness depend, we will con 
sider the wider problem of relations in general possess 
ing these two properties. There is one mode of con 
structing such relations which has very wide application 
and is of great importance in logical theory, viz. 

*x is r to the thing that is r to z 

Here the word the indicates that r is a many-one rela 
tion. I shall call the thing to which reference is made 
in the above formula the intermediary term, and the 
relation r the generating relation. Thus, given an in 
termediary term and a many-one generating relation, 
we can always construct by (what is called) relative 

J. L. II 10 



146 CHAPTER VI 

multiplication a derived relation which is symmetrical 
and transitive. Representing the intermediary by the 
symbol y, the relation of x to z may be otherwise ex 
pressed by the conjunctive proposition : 

x is r to y and z is r to jj// 

where it is to be understood that there is some uniquely 
determined entity (say j/) to which ^ and 2 stand in the 
relation r\ i.e. r is a many-one relation. 

Now the theorem that any relation so constructed 
is symmetrical and transitive requires no discussion 
and is universally admitted ; but the converse theorem 
--that any symmetrical and transitive relation can be 
exhibited by this mode of construction cannot be 
assumed to be true without careful examination. To 
this converse theorem Mr Russell gives the name the 
principle of abstraction ; and professes to have proved 
its truth by a process involving highly complicated 
symbolism. It is quite easy, however, to explain the 
nature of his proof without recourse to such symbolism. 
Thus, let f be a symmetrical and transitive relation ; 
then, in order to prove the theorem, we have to dis 
cover an intermediary entity and a generating relation 
in terms of which t may be constructed. The inter 
mediary entity for the relational predication x is t to 2 
is, in Mr Russell s proof, the class of things comprising 
x together with everything such as z for which "x is t to 
z" holds. The required generating relation r is the rela 
tion of being comprised in ; hence the proposition x is 
t to z 9 is resolved into the form: 

x is comprised in the class 

(defined as comprising everything to which x is ?) 
which comprises z. 



FUNCTIONAL DEDUCTION 147 

Here the intermediary entity is a class uniquely defined 
in terms of x and t, and therefore the relation in which 
x or any other item stands to the intermediary is a 
many-one relation. Now what Mr Russell has suc 
ceeded in proving in this way is proved with absolutely 
demonstrative validity; but my first comment is : has 
he proved what he undertook to prove ? In one sense 
he has proved too much, and in another sense he has 
proved nothing whatever that is relevant. He has 
proved too much in the sense that he has discovered an 
intermediary entity which would, mutatis mutandis, 
apply to every possible symmetrical and transitive rela 
tion, such as contemporaneous, compatriot, co-implicant, 
co-incident, as well as equal. Thus he has proved that, 
for the resolution of the relation equals, we must take as 
intermediary the class of quantities equal to any given 
quantity ; for the relation contemporaneous, the class 
of events contemporaneous with any given event ; 
for the relation compatriot, the class of persons that 
are compatriots of any given person ; and so on. But 
what he set out to discover as the required inter 
mediary was, in the case of equality, a certain magni 
tude-, in the case of compatriot, a certain country, in 
the case of contemporaneous, a certain date , and so 
on. He has not proved that there is a certain magni 
tude that all equal quantities possess ; nor a certain 
country to which all compatriots belong; nor a certain 
date to which all contemporaneous events are to be 
referred. Moreover, in taking as his intermediary a 
certain uniquely determined class, it seems obvious that 
Mr Russell s alleged proof is incomplete, unless we can 
assert that there are such entities as classes, and the 



102 



148 CHAPTER VI 

validity of this assertion is explicitly denied by him : or 
rather he holds that there is no necessity in the deduc 
tions of logic and mathematics to assume that there are 
classes, although without this assumption his proof of 
the principle of abstraction completely breaks down. 

I do not, however, wish to press my criticism of 
Mr Russell further, but rather to expound what ap 
pears to me to be the true view on the nature of 
abstraction. The cases in which the principle comes 
into consideration may be distinguished according as 
the intermediary is of the nature of a substantive such 
as country, or of the nature of an adjective such as 
magnitude. In applying the attempted proof of the 
principle of abstraction to such a relation as compatriot, 
Mr Russell argues as if we knew this relation to 
be symmetrical and transitive independently of our 
knowledge that a person can belong-to (r) only one 
country (jy); whereas it is obvious that we have con 
structed the derivative relation compatriot by means of 
the prior notions country and belonging- to. Hence, no 
such case as compatriot can be used to prove the prin 
ciple of abstraction, but only to illustrate the theorem 
of which the principle of abstraction is the converse. 
Where the intermediary is adjectival, e.g. colour, pitch, 
magnitude, the principle directly raises the issue of the 
connection and distinction between a determining ad 
jective and the class that it determines. In the case of 
an adjectival intermediary, our general formula 

x is r to the term (say y) that is r to z* 

must be expressed in a special form in which the gene 
rating relation (r) is to stand for characterised-by (x), 



FUNCTIONAL DEDUCTION 149 

and the intermediary term (y) is to stand for a specific 
determinate under a specific determinable, thus : 

x is characterised-by the determinate adjective 
that characterises 2. 

Here the uniqueness of the intermediary term is secured 
by the disjunctive principle of adjectival determination 
expressed (Part I, Chapter XIV) in the form: Nothing 
can be characterised by more than one determinate 
under any assigned determinable. Now, since any one 
substantive may be characterised under many different 
determinates, the intermediary term must specify the 
determinable, or ground of comparison, upon which the 
symmetry and transitiveness of the derived relation 
depend. Thus, 

*x is characterised by the colour that characterises , 
or x is characterised by the shape that characterises , 
or x is characterised by the size that characterises z* 

Any of these three propositions may be significantly 
asserted of the same subjects x and z, if these are 
bounded surfaces distinguished from one another by 
determinate localisation ; and the relation of x to z thus 
constructed is transitive (as well as symmetrical) pro 
vided that the colour, shape or size is strictly deter 
minate. With this proviso, we may say that x and z. 
are equivalently coloured, equivalently shaped or equi- 
valently sized, as the case may be. Such symmetrical 
and transitive relations between the substantives x and 
z must be distinguished from the symmetrical and tran 
sitive relation identity which holds between the adjectives 
described as the colour of x and the colour of z, the shape 
of x and the shape of 2-, or the size of x and the size of z. 



150 CHAPTER VI 

Now magnitude like any other adjectival determinable 
must first be abstracted as a character in order that by 
its means we can construct the class of equally sized 
objects. Thus it is just as absurd to define the size of 
x in terms of the class of objects that are equal in size 
to x as to define the colour of x in terms of the class 
of objects that are equivalent in colour to x! 

12. To secure that the relations constructed by 
means of the above formula shall be symmetrical and 
transitive, it is necessary to specify, not only such differ 
ences as those between colour, shape, etc., but also 
differences within the general notion magnitude, con 
stituting various kinds or species of magnitude. For 
just as colours and sounds are incomparable with one 
another, since they must be characterised under dif 
ferent determinables, so there are distinct determinables 
subsumable under the superdeterminable magnitude. 
Taking some of Mr Russell s suggestive examples, we 
note that the magnitude of pleasure predicable of an 
experience is incomparable with the magnitude of area 
predicable of a surface, and that these again are in 
comparable with the magnitude of duration predicable 
of an event. Hence pleasure-magnitude, area-mag 
nitude, duration-magnitude, are three distinct deter 
minables, predicable only of experiences, surfaces, and 
events respectively. In ordinary usage the word 
magnitude is omitted when reference is made to the 
determinables in question ; but in specifying the area 
of a surface, we are in point of fact specifying a kind 
of magnitude; so in specifying the duration of an 
event we are specifying another kind of magnitude; 
and in specifying the pleasure of an experience, we 
are specifying yet another kind of magnitude. The 



FUNCTIONAL DEDUCTION 151 

analogy here drawn between area or duration on the 
one hand, and pleasure on the other will probably be 
disputed because pleasure is so often used in its concrete 
sense to mean pleasurable experience as well as in its 
abstract sense to mean * the pleasure of a (pleasurable) 
experience. Now it happens that a pleasurable ex 
perience may be characterised under at least two dif 
ferent determinables of magnitude; viz. pleasure-mag 
nitude and duration-magnitude, the latter of which 
applies in the same sense to any event whatever that 
may last through a period of time. Here it is important 
to note that pleasure-magnitude and duration-magni 
tude, etc. are not determinates under the one deter- 
minable magnitude, but different species included in 
the genus magnitude. They may therefore be con 
veniently termed sub-determinables of magnitude, 
each generating its own determinates, which are in 
comparable with the determinates generated by any 
other. Thus magnitude does not generate its sub- 
determinables in the way in which a determinable gene 
rates its determinates. An experience, a surface, an 
event are substantives belonging to different categories 
of which pleasure, area, or duration may be respec 
tively predicated as adjectives ; but a specific pleasure- 
magnitude, or area-magnitude, or duration-magnitude 
is related to its respective species of magnitude as a 
determinate to its determinable. We shall proceed in 
the next chapter to examine and classify the fundamental 
kinds of magnitude, to which reference is here made. 

13. It remains to point out one highly important 
characteristic which distinguishes pure or pre-mathe- 
matical logic from mathematics proper. I n both branches, /! 
the two principles of inference termed Applicative and 



I 5 2 CHAPTER VI 

Implicative are employed in the procedure of functional 
/inference, and these alone. But the peculiarity of 
pre-mathematical deduction is that it lays down two 
formulaeofimplication(e\ti\etzs primitive or as derived) 
which are virtually equivalent respectively to the Appli 
cative and Implicative Principles themselves. The 
formulae in question may be thus expressed : 

(1) Applicative formula . Any predication that holds 
for every case x would formally imply that the same 
predication holds for a given case a. 

(2) Implicative formula : For any case x, y, the com 
pound u jtr" and "x would imply jx" would formally 
imply y* 

We must, therefore, explain the distinction between 
Principles of Inference, on the one hand, and Formulae 
of Implication, on the other hand. In all formulae of 
implication, the implicans and implicate stand indif 
ferently for propositions that are to be materially or 
formally certified. But, when a formula of implication 
is used as a premiss in the process of deduction, its 
implicans must first be formally certified in order that 
its implicate may be formally certified. This inference 
is made by a direct application of the implicative prin 
ciple. And again, every formula of implication holds 
for all cases coming under an assigned form ; hence the 
inferences from any formula of implication are made by 
a direct application of the applicative principle. The fact 
that every step by which we advance in the building 
jup of the logical calculus requires both the Applicative 
>and the Implicative principles of inference, and these 
alone, establishes their sovereignty over all deductive 
; processes. 



CHAPTER VII 

THE DIFFERENT KINDS OF MAGNITUDE 

i. THE term magnitude, as is suggested by its 
etymology, denotes anything of which the relations 
greater or less can be predicated; and it is only if M 
and N (say) are magnitudes of the same kind that M 
can be said to be greater or less than N. I have taken 
magnitude to be an adjectival determinable, or rather a 
class of adjectival determinables including several dis 
tinct kinds. That of which a determinate magnitude 
of a specific kind may be predicated stands, relatively 
to its magnitude, as substantive to adjective ; but it may 
be either an existent, i.e. substantive proper (in which 
case the magnitude predicated is a primary adjective) 
or itself an adjective (in which case the magnitude pre 
dicated is a secondary adjective). In order to keep 
clear the distinction between the adjectives of magni 
tude themselves and the substantives of which magni 
tude is predicable, a separate terminology ought strictly 
to be applied to the latter. A striking case where 
language supplies us with the logically required termi 
nological distinction is that of longer and shorter 
predicated of lines to the lengths of which the terms 
greater and less are applied. 1 1 would be convenient, 
for the purposes of a general exposition of magnitude, 
to restrict the application of the terms greater and 
less to magnitudes, and to adopt the corresponding 
terms larger and smaller for that of which the 



I54 CHAPTER VII 

magnitudes are predicated. For example: the class 
compositae is larger or smaller than the class violaceae, 
according as the number of compositae is greater or less 
than the number of violaceae 1 ; the period 1815 to 1832 
may be called larger than the period 1714 to 1720, 
inasmuch as the temporal magnitude of the former is 
greater than that of the latter. Now for every distinct 
kind of magnitude there is a corresponding distinct kind 
or category of entity of which it can be predicated ; and 
hence, though it is strictly illogical, yet it is legitimate 
and usual to apply the same terms, such as extensive 
and intensive, to distinguish both between the different 
kinds of magnitude and between the corresponding 
different kinds of entities which bear to the magnitude 
the relation of substantive to adjective. From these 
preliminary remarks, we may pass to an examination of 
the nature of different kinds of magnitude, beginning 
with number, which is the most fundamental of all. 

2. Integral number is an adjective exclusively 
predicable of what we call classes, including enumera 
tions; two classes being said to be numerically equal 
when the number predicable of the one is identical with 
that predicable of the other. I think it is legitimate to 
maintain that the two notions class and number are not 
independently definable, but each definable only in its 
relation, the one as the only appropriate substantive for 

1 This may mean either that the number of existing plants com 
prised in the genus is greater or less, or that the number of infimae 
species included in the genus is greater or less. It is obvious that these 
two modes of determining numerical comparison do not necessarily 
tally. It will be shown later that the same distinction holds as regards 
the number of points in a line and the number of linear parts (equal 
or unequal) into which it may be exhaustively and exclusively divided. 



THE DIFFERENT KINDS OF MAGNITUDE 155 

the other as its only appropriate adjective. The common 
habit of representing classes by closed figures may lead 
to the false supposition that the members of a class 
can as such be arranged in some kind of proximity to 
one another within an enclosed space. But when the 
items to be comprised in a class have relations meta 
phorically called near or far, they constitute not merely 
a class but a series or ordered class. Now in modern 
mathematics the appropriate number-adjective of a 
class conceived independently of any arrangement or 
order of its items, is known as a cardinal number; 
whereas of a series or ordered aggregate the appropriate 
numerical adjective is known as an ordinal number. 
When a class or enumeration comprises a finite number 
of items, then, in whatever order the items may be 
enumerated, we reach the same ordinal number, and 
this number agrees with the cardinal number; but for 
transfinite aggregates, which have been introduced into 
modern arithmetic, this agreement no longer holds; 
and consequently the fundamental distinction between 
ordinal and cardinal numbers is required. Readers are 
referred particularly to Mr Russell s Principles of Mathe 
matics for the full development of this topic, which is 
outside the compass of my work. 

3. The psychological aspect of number is revealed 
by analysing the process of counting. In this process 
we establish numerical equality between a set of things, 
on the one hand, and a set of number-names temporarily 
attached to the things, on the other hand. Hence 
counting is a special, and, in some respects, a unique 
case of correlation between the things upon which names 
are imposed and the names that are imposed upon the 



I 5 6 CHAPTER VII 

things. Ideally language requires that any given proper 
name should denominate one and only one thing, and 
conversely that any given thing should be denominated 
by one and only one proper name ; or briefly, that there 
should be a one-one correlation between the names of 
things and the things named. If this relation held, it 
would follow that the class of names would be numeri 
cally equal to the class of things named. Actually, 
however, this ideal is not realised; for the same thing 
often has many names, and the same name is often 
attached to many things. It is worth pointing out that 
there may still be numerical equality in spite of there 
not being a one-one correlation between names and 
things named. For example : let R, Q, M, T, U be a 
set of names, and /c, cr, 0, x> <l> a set f things named. 
Then suppose that 

R names K or cr ; Q names K or cr ; M names cr or 
6 or x ; 2" names ^ ; and U names 6 or < or K or ^ ; 
so that 

K is named R or Q or U; <r is named R or Q or 
M\ 6 is named M or U] x ls named M or Tor 
U\ and <j> is named U. 

Here the denominating correlation is not one-one but 
many-many, and yet the names and the things happen 
to be numerically equal. How then do we establish the 
fact that the number of items in the enumeration R, Q, 
M, T y U is the same as that in the enumeration /c, cr, 
6 y x> <? What we do, where there is no factual correla 
tion, is to institute what I shall call a factitious corre 
lation ; by which I mean one which is not inherent or 
objective, but arbitrarily imposed by the counter. In the 
adduced instance in order to establish the numerical 



THE DIFFERENT KINDS OF MAGNITUDE 157 

equality between the enumerations R, Q, M, T, /, 
and /c, cr, 0, ^, <j> we must mentally attach, either in 
thought or in figurative imagery, R say to 9, Q to <, 
T to 0-, U to K y M to x J where the items of the two 
sets have been indiscriminately permuted and attached. 
We can now analyse the mental act of counting as a 
special case of factitious correlation. The essential 
psychological requisite is that we should learn to enu 
merate a set of arbitrary names in a foxed or invariable 
temporal order from the first onwards ; and these names 
are attached temporarily to the objects to be counted, 
in this respect differing from names in general which 
have fixed denotation. For example: let us arrange 
the names U, R, Q, M, T in the following order: 
M y Q, R, 7", U\ and temporarily attach these names 
as follows : M to x> Q to <t>> & to #> T to o-, U to K. 
Thus the set of names have to be attached in a fixed 
order, one by one, to the set of things taken in any 
order. What is logically required to avoid mistake is 
that the enumeration of the things should be both ex 
haustive and non-repetitive a condition which children 
and savages often find difficult to fulfil. Now, inasmuch 
as the number-names M, Q, R, T, /are always attached 
in an invariable order, the last number named indicates 
unequivocally the number of the counted set of objects. 
In other words, the cardinal number of any enumerable 
set of objects is unambiguously indicated by the ordinal 
number of the correlated number-names. Historically 
the letters of the alphabet, having been memorised in 
a fixed order, served also as the written symbols for 
numbers; but their employment for this purpose could 
not be extended to all numbers, since an alphabet 



158 CHAPTER VII 

necessarily consists of a limited number of letters. 
Moreover it is psychologically impossible to memorise 
an endless list of names. Hence it was necessary to 
invent some system which would render it possible to 
count any set of things, however large. The Roman, 
Greek and Hebrew alphabets were employed for this 
purpose with more or less success, but were finally 
superseded by the Arabic notation in which place-value 
was given to the symbols i, 2, 3, 4, 5, 6, 7, 8, 9, and 
the symbol o was added. These ten symbols serve as 
proper names of numbers, all other numbers being ex 
pressed by names constructed out of these. Thus the 
compound word twenty-four or the compound symbol 
24 is analysable as meaning two tens plus four, and 
therefore to be understood in terms of the operations 
of multiplication and addition. Such compound symbols 
or words are not proper names of numbers like two, ten, 
or four, but may be called constructed names. The 
elementary learner of arithmetic must, in fact, reverse 
the logical order of thought, and understand the pro 
cesses of multiplication and addition before he can 
intelligently learn to count beyond twenty or so, or 
understand what is known as the decimal system of 
notation. 

4. We now pass from the psychological analysis of 
counting to the consideration of its underlying logical 
principles. Counting is a special case of one-one corre 
lation, the peculiar characteristics of which are (i) that 
a prescribed set of name-items have to be memorised 
in a definite serial order; and (2) that the correlations 
are factitious. As regards (i) the mental process of 
counting, which involves order, must be contrasted with 



THE DIFFERENT KINDS OF MAGNITUDE 159 

one-one correlations in general which are irrespective 
of order. As regards (2) factitious correlations must be 
contrasted with such correlations as husband and wife, 
denominating and denominated by (in an ideal language) 
etc., in which any given husband is correlated with a 
determinate wife, or any given proper name with a 
determinate thing. Now in my view, factitious corre 
lations are essentially necessary in the general theory of 
numerical equality; though they never enter into the 
abstract deductions of arithmetic. On the necessity of 
factitious correlations, recognised authorities, above all 
Mr Russell, are opposed to me. Their definition of the 
numerical equality of two sets of things is, in effect, 
formulated as follows: There is a one-one relation of 
any member of the one set to some member of the 
other set. But it seems to me essential to distinguish 
the statement that the items can be correlated one 
to one from the statement there is a one-one correla 
tion ; the former points to a factitious, the latter to a 
factual correlation. There need be no relation at all de 
pending on the nature of the items themselves comprised 
in the two sets, that would determine which item of the 
one set should be attached to any given item in the 
other. If relations are treated extensionally, i.e. as mere 
substantive-couples, then it is of course a matter of 
fact that two numerically equal classes contain couples 
of items, one of which is comprised in the one class and 
the other in the other; but I know of no sense in which 
the two members of the couple are related the one to 
the other, except that the one is temporarily attached 
in thought by some thinker to the other. Apart from 
this factitious coupling, there is no one-one relation, 



160 CHAPTER VII 

subsisting between any given item of the first set and any 
determinate item of the second, that would not equally 
subsist between the given item and any other item arbi 
trarily selected from the second set. Thus the establish 
ment of numerical equality between two finite classes 
requires in general factitious correlations. On the other 
hand, the only mode of establishing numerical equality 
between infinite classes is to discover factual, or more 
specifically formal, correlations. The formal correla 
tions required in pure arithmetic, finite and transfinite, 
are what may be called functional ; and, for the purposes 
of this elementary exposition of the logic of arithmetic, 
the notion of functional correlation must be introduced 
and explained. 

5. Using the symbol f for any function and/" for 
its converse, the relation n tof(n) will be one-one; pro 
vided that n determines uniquely the value otf(n) and 
f(m) determines uniquely the value of f{f(vt)} For 
example: let/(^) stand for # + 7, then f(m) will stand 
for m 7 ; and the integers from i to n (inclusive) can 
be correlated one to one with the integers from 8 to 
n 4- 7 ; each integer in the second series being given by 
adding 7 to the corresponding integer in the first, and 
each in the first series by subtracting 7 from the corre 
sponding integer in the second. Similarly, \if(n) stands 
for n x 7, then/(m) will stand form + j; and the integers 
from i to n can be correlated one to one with the multi 
ples of 7 from 7 to 7#. In general : if the relation of n to 
f(n) is one-one, then the series of values assumed by n is 
numerically equal to the series of values assumed by/(n). 

An important application of this theorem is to the 
case where the integer n assumes all possible finite 



THE DIFFERENT KINDS OF MAGNITUDE 161 

values, obtained from unity by the successive addition 
of unity. In this case, the simplest illustration is afforded 
by taking f(n) to stand for zn. There is then established 
a one-one correlation of the successive integers i, 2, 
3, 4 ... with the successive integers 2, 4, 6, 8 In 
other words, the number of finite integers is the same 
as the number of finite even integers; although the 
former series comprises all the odd integers and these 
are not comprised in the latter. Thus, although the 
aggregate of even integers is a part proper of or sub- 
included in the aggregate of integers, yet the two 
aggregates are numerically equal. Now we may define 
an infinite number as the number of any aggregate that 
includes a part proper numerically equal to itself. Thus 
the instance above cited is the simplest of the many 
proofs that establish the theorem that the number of 
finite integers is infinite. If the integers are presented 
in ascending order of magnitude, the series so conceived 
has a first but no last term and also is discrete in the 
sense that each term has one and only one immediate 
successor. The cardinal number of any aggregate that 
can be so arranged in a series is called X . This is the 
smallest of infinite cardinal numbers. 

The reader must here be referred to the mathe 
matical exponents of the theory of transfinite cardinals 
and ordinals for further instruction. The most com 
prehensive account of this theory will be found in 
Mr Bertrand Russell s work entitled Principles of 
Mathematics. 

6. As number and the magnitudes that are derived 
solely from number may be called abstract, so those 
which contain a material factor may be called concrete 

J.L.II II 



162 CHAPTER VII 

magnitudes or quantities. Thus duration-magnitudes, 
stretch-magnitudes, magnitudes of qualitative difference 
are quantities, because the entities of which they are 
predicable are defined and differentiated in terms that 
are not purely logical. This use of the term quantity 
differs from that expressly enjoined by Mr Russell, who 
defines a quantity as an instance or specification of 
magnitude. He then proceeds to identify the relation 
thus indicated in some cases with that of substantive to 
adjective, and in others with that of determinate to de- 
terminable; whereas, in the common language of mathe 
matics, quantity stands to magnitude in the relation of 
species to genus, with which my use of the term quan 
tity corresponds. With regard to quantities the three 
differentiae which I hold to be fundamental or primitive 
are extensive, distensive and intensive. The term dis- 
tensive magnitude is new, and the reason for placing it 
intermediarily between extensive and intensive is that 
by some logicians it has been included under extensive 
and by others under intensive magnitude. 

An extensive magnitude may be defined as one 
which can be predicated only of an entity that can 
appropriately be called a whole. The notion of whole 
is correlative to the notion of part; and, more precisely, 
a whole is to be conceived as having parts which can 
be specifically identified and distinguished indepen 
dently of their relations of equality or inequality ; e.g. 
a finite line is a whole of the simplest possible kind, 
under the figure of which all one-dimensional wholes 
may be metaphorically pictured. Thus a line CEG is 
represented as having the parts CE and EG, each of 
which is definitely identifiable for itself and distinguish- 



THE DIFFERENT KINDS OF MAGNITUDE 163 

able from the other. The several parts of a whole are 
of the same nature as the whole, and therefore the con 
struction of a whole out of parts or the division of a 
whole into parts may always be called homogeneous 1 . 
The term extensive magnitude has, in fact, been popu 
larly restricted to spatial and temporal wholes; but I 
shall follow Mr Russell in applying this term also to 
certain qualitative wholes, e.g. to a continuous aggre 
gate of hues or of pitches. Thus we speak of a scale of 
hue and a scale of pitch in the sense of a class com 
prising all specific hues or pitches which are qualita 
tively intermediate between two terminal hues or pitches. 
Now the class comprising such determinate items con 
stitutes what is now called a stretch ; thus a qualitative 
stretch of hue or of pitch is formally analogous to the 
period comprising all determinate instants between one 
instant and another or to the geometrical line comprising 
all points intermediate between one point and another. 
7. It might appear, since the instants comprised 
in a period and the points comprised in a line are sub 
stantival, while the hues or pitches comprised in a 
qualitative stretch are adjectival, that there is some 
fundamental logical distinction between these two kinds 
of stretches. Thus : though either may be metaphori 
cally represented by a line CEG, yet, if the points C, 
E, G stand actually for points or instants these being 
substantival the stretch represented is substantival ; 
whereas if C, E, G represent three pitches pitches 

1 The term whole is frequently applied to a construct constituted 
of heterogeneous elements, e.g. to a proposition ; but for such a con 
struct the term unity is preferable, unity being the genus of which 
whole is a species. 



II 2 



164 CHAPTER VII 

being characteristics of sound and therefore adjec 
tival the stretch itself is adjectival. It is open to 
question, however, whether points or instants are of a 
substantival nature ; and this has been a matter of 
frequent philosophical dispute. If we regard time and 
space as existents, then the events which occur at a 
given date or occupy a given period, like the ink-spots 
which may be placed at different points or the ink- 
lines which may be drawn on paper, have as substan 
tives a unique kind of relation to the substantives of 
a different category date, period, point or line to 
which they are attached. Such a relation, like that of 
characterisation, is unique ; in the sense that one of 
its terms necessarily belongs to a certain category and 
the other to a certain other category. The relations 
occupying J and occurring at further resemble the 
characterising tie in being unmodifiable ; thus, of any 
given date and any given event the only relevant as 
sertion that can be made is that the event either did 
or did not occur at that date. In saying of an event 
that it occurs at a certain date or of a material body 
that it occupies a certain region, the predications may 
be, not ultimately analysable into definable relations 
to a definable period or region, but regarded rather 
as adjectivally unanalysable. Occurring at and occu 
pying are therefore properly speaking ties. It is not, 
however, formally incorrect to regard them as relations, 
in the same way as we have allowed characterisation to 
be analysed as a relation involving the two correlatives 
characterising and characterised by. From this discus 
sion it will be seen that I incline to the view that 
instants of time and points of space, as well as time 



THE DIFFERENT KINDS OF MAGNITUDE 165 

and space as wholes, are not substantival or existential 
but merely adjectival. 

Now, under the head of the relativity of time and 
space two distinct philosophical problems are often 
confused. The view that position in space or time is 
definable not as absolute, but only as relative to other 
points or instants is to be distinguished from another 
view according to which temporal and spatial relations 
are relations, not between entities such as points and 
instants, but between what occupies the points or in 
stants. The first of these two problems is appropriately 
described as the question of the absoluteness or rela 
tivity of time and space ; the second as the question of 
the substantival or adjectival nature of time and space. 
In the Principles of Mathematics Mr Russell explicitly 
maintains the absolute view as regards both these 
problems ; he deliberately asserts that position a term 
conveniently used both for space and time is absolute 
and not merely relationally definable in terms of other 
points or instants ; and also that points and instants 
are existents. Now, in the foregoing analysis, I have 
taken the relative i.e. adjectival view on the second of 
these two problems, while not rejecting the absolute 
view on the first. The adjectival view of space and 
time, in which we deny such separable entities as 
instants and points, must not be confounded with the 
class-view : that identity of dating merely means being 
comprised in a certain assigned class of contempora 
neous events. For, in holding that occupying a certain 
instant is an unanalysable adjectival predicate, we 
maintain at the same time that, qua predicate, it is an 
identifiable entity, in the same way as the adjective 



166 CHAPTER VII 

red is an identifiable entity when predicated now of 
this patch and then again of some existentially other 
patch. This view is not inconsistent with my previous 
analysis ; for I have repeatedly maintained, particularly 
in my analysis of the principle of abstraction, that ad 
jectival identity cannot be resolved merely into mem 
bership of a certain definable class. My contention for 
the adjectival nature of space and time amounts to the 
statement that instants and points are substantival 
myths. It is not necessary, however, for the purposes 
of this exposition, to press the question of the substan 
tiality of time and space, for any difference of view on 
this point does not affect the further development of 
the subject. 

8. Having shown the analogies between the three 
kinds of stretches qualitative, temporal, and spatial 
or rather linear we will now compare such extensive 
wholes with classes considered in extension, which may 
be called extensional wholes. It is not a mere accident 
of language that the term extension has two applica 
tions in philosophy, these generally occurring in such 
different contexts that they are not confused. But it is 
worth while drawing attention to the double use of the 
word ; and, in so doing, to examine a topic, prominent 
in modern mathematics, concerning the formal agree 
ments and differences between extensional and exten 
sive wholes. An extensional whole, otherwise a class, 
is naturally associated with the notion of assignable 
items of which the class is composed ; on the other hand, 
a linear whole which illustrates an extensive whole of 
the simplest kind is apprehended as a whole without 
thinking of the points it contains. In other words : 



THE DIFFERENT KINDS OF MAGNITUDE 167 

the items in an extensional whole are prior in thought 
to the whole, which appears to be the product of a 
constructive process ; while the extensive whole is 
prior to any conception of points, which seem to be the 
result of a similar, but reversed, process of thought-con 
struction. This psychological distinction Mr Russell 
seems to regard as philosophically negligible ; and he 
devotes a large part of his exposition to a proof of the 
essential sameness in nature of extensional and ex 
tensive wholes. This question raises the same problem 
as that discussed by Hume and Kant the former in 
his quarrel with the mathematicians, and the latter in 
his solution of the antinomies. 

Let us then examine what common-sense would elicit 
from a consideration of these two kinds of wholes. With 
regard to extensional wholes, I have adopted the term 
comprise to represent the relation of a class to any 
of its items or members, and include to represent the 
relation of a genus to any of its species ; and it is of 
the first importance to note that, for extensive wholes, 
an analogous distinction holds between the relation of 
a line to any of its points and the relation of a line to 
any of its parts which are themselves linear. For just 
as a class comprises items which have to one another 
the sole relationship of otherness, so a line comprises 
points which have to one another the sole relationship 
of otherness ; and again, just as members of a species 
are members of the genus, so points in a linear part 
are points in the linear whole. Further, since a line or 
stretch contains parts in the same sense as a genus 
includes species, it follows that such purely logical or 
formal relations as overlapping, includent, excludent, 



168 CHAPTER VII 

which apply to classes and not to items, apply also to 
the parts of a stretch but not to points. Now the parts 
into which a three-dimensional space can be divided 
are three-dimensional, and have, qua three-dimensional, 
all the properties of the whole ; similarly the parts of 
a two-dimensional space are two-dimensional ; and the 
parts of a one-dimensional space one-dimensional. On 
the other hand, of the contiguous parts of a three- 
dimensional whole the common boundary is two-dimen 
sional ; of the contiguous parts of a two-dimensional 
whole the common boundary is one-dimensional ; and 
of the contiguous parts of a one-dimensional whole 
the common boundary is zero-dimensional, i.e. a point. 
Restricting our discussion to the last case, we note a 
very substantial difference between extensive wholes 
and extensional wholes ; for within a merely extensional 
whole there are no relations of contiguity, whereas 
every extensive whole is apprehended as containing 
parts which are either literally or metaphorically further 
from or nearer to one another. Hence the notion of a 
point as a boundary comprised in neither or in both of 
the two parts of a line has no analogy amongst members 
of a genus which belong either to one species or to 
another and cannot belong to both. It further follows 
that an extensive whole resembles a serial or ordered 
set of items rather than a mere unordered class or 
enumeration. 

9. Having considered the nature of an extensive 
whole, i.e. that of which extensive magnitude may be 
predicated, we will pass to the consideration of the 
kinds of entities of which distensive or intensive mag 
nitude can be predicated. By distensive magnitude is 



THE DIFFERENT KINDS OF MAGNITUDE 169 

meant degree of difference, more particularly between 
distinguishable qualities ranged under the same deter- 
minable 1 . Thus the difference between red and yellow 
may be greater or less than that between green and 
blue ; and similarly the difference between the pitches 
C and F may be greater or less than that between B 
and G. The notion of difference is apt to be associated 
with the arithmetical process of addition, for which the 
term addendum or subtrahend may be used in order 
to distinguish it from a distensive magnitude. Thus it 
is preferable to say that successive terms forming an 
arithmetical progression are obtained by a constant 
addendum, just as those forming a geometrical pro 
gression are obtained by a constant multiplier. This 
reference to arithmetical and geometrical progressions 
is needed because the measure of qualitative difference, 
in its logical and even its philosophical sense, is in 
some cases or on some grounds to be conceived as an 
addendum, and in other cases or on other grounds as a 
multiplier. For example, if a series of colours are pre 
sented as in the spectrum, in a continuous spatial order, 
we might conceive the magnitude of difference between 
any one hue and any other to be proportional to the 
length in the spectrum between the two hues. In this 
case, by taking any hue as origin, say O, such that A 
is between O and B, and representing the difference 
between A and B by the symbol AB, we should assume 
that its value was given by the equation AB = OB OA. 
On the other hand, as regards the scale of pitch, the 
scientist would naturally connect the pitches with the 
physical process of aerial vibration, and measure each 
1 See Part I, p. 191. 



170 CHAPTER VII 

pitch by the number of vibrations per second. On this 
assumption the difference between C and G would 
be represented by the ratio of f, and that between G 
and B by , and therefore the difference between C 
and B would be f x % = /-. These two examples of the 
two natural modes of estimating degrees of qualitative 
difference viz. by an addendum or by a multiplier 
are typical of all problems regarding distensive or even 
intensive magnitudes. 

It will be important, however, to contrast either of 
these more physical modes of conceiving distensive 
magnitude with the mode that has become familiar to 
psychologists ever since Fechner s and Weber s experi 
ments. According to Fechner it would appear that the 
magnitude of difference between the qualities or inten 
sities of sensations should be determined by taking as 
unit-difference that which is just discernible in an act 
of perception directed to the sensations as experienced. 
It should be here noted that we are measuring psychical 
entities, and not, as in the previous discussion, their 
physical correlates. Fechner adopted the view that the 
proper sensational magnitude, either of qualitative or 
of intensive difference, was obtained by addition, in 
which equal units were those which were just per 
ceptible. When he compared the resulting sensational 
magnitude with the magnitude of the stimulus as 
measured physically, he concluded that, while the sensa 
tions could be ranged in arithmetical progression, the 
corresponding stimuli would form a geometrical pro 
gression. By means of an elementary mathematical 
process it will be seen that this formula can be ex 
pressed by saying that the magnitude of the sensation 



THE DIFFERENT KINDS OF MAGNITUDE 171 

varies as the logarithm of the stimulus. But this tech 
nical development is not our concern here. I wish 
rather to draw attention to the extraordinary, and in 
my view baseless, assumption that the just discernible 
differences at the different points in a scale should be 
taken to indicate equal addenda. If he had assumed 
what appears to be more plausible that the just dis- 
criminable qualities were those which bore a common 
ratio to one another, the experimental results of Fechner 
or Weber would have been most naturally expressed 
in the formula that the sensational magnitude (or) varies 
in proportion to the magnitude of the physical stimulus 
(s) measured from a certain constant (s ) : i.e. 

<r = 6(s-s ), 
where k is constant; instead of by the formula 



So far from taking discriminability as equivalent to an 
addendum, it is more plausible to consider it as equiva 
lent to a ratio 1 . For example, taking the visual magni 
tudes of four objects A t B, C, and D, if we can just 
discriminate between the magnitudes of A and B and 
also between those of C and D, then it is reasonable 
to infer that the ratio of B to A is equal to the ratio 
of D to C, rather than that the addendum by which 
B exceeds A is equal to the addendum by which D 

1 I am not here concerned with the accuracy of the experiments 
made by Fechner, nor with his right to make the very wide induction 
from the artificial nature and limited number of cases that he and 
his successors have examined. I am referring merely to a logical and 
not to a psychological question, namely, the justification for regarding 
our power of discrimination as equivalent to our power of perceiving 
additions of magnitude rather than ratios. 



172 CHAPTER VII 

exceeds C. This principle for measuring psychical 
magnitude may be applied not only to cases of direct 
sense-perception, but also to those in which we are 
guided by general psychological considerations ; for 
example, the difference of pleasure that we conceive 
to be produced by two different increases of income 
such as that from ,100 to ^200 and from ^1000 to 
^noo would not naturally be taken to be equal ; the 
increase from ^100 to i 10 would rather be considered 
the equivalent of the increase from ^1000 to ^noo. 

10. We now pass explicitly to the third funda 
mental kind of magnitude, namely intensive, which has 
received considerable philosophical attention. Kant 
regarded intensity as so to speak equivalent to existence 
or reality, so that that which has greater intensity has 
for him greater reality. The point in which this view 
agrees with the modern theory is that intensity has a 
terminus in the value called zero ; and it is in this 
respect that the distinction between distensive and 
intensive magnitudes is most clearly marked ; the mini 
mum or zero of distensive magnitude is identity, whereas 
the minimum or zero of intensive magnitude is non- 
existence. Another obvious distinction between the two 
kinds of magnitude is that distensive magnitude is a 
relation between determinates under some one given 
determinable, whereas intensive magnitude holds within 
each separate determinate, or even amongst different 
qualities under the same determinable. Thus, with re 
gard to the comparative brightness of different hues, 
we may predicate equal to, greater than or less than, 
and so also with regard to the loudness of sounds of 
different pitch. It is impossible, however, to compare 



THE DIFFERENT KINDS OF MAGNITUDE 173 

two kinds of intensive magnitude such as the bright 
ness of a light sensation with the loudness of a sound 
sensation ; all we can say is that a colour of zero 
brightness would be non-existent, and a sound of zero 
loudness would be non-existent. The subtle point then 
arises whether the notion of zero-intensity of sound is 
distinguishable from the notion of zero-intensity of 
light. In popular language we might ask: Is there 
anything to distinguish absolute silence from absolute 
darkness ? I think that apart from an organ of sensa 
tion having potentialities as a medium for receiving 
sensations we must say that zero-intensities are indis 
tinguishable ; it is only through the capacity of visual 
and auditory imagery, and indirectly through the pos 
session of organs for conveying these two corresponding 
kinds of sensation, that distinctions between zeros can 
have for us any import. 

ii. In conclusion I have to explain why disten- 
sive magnitudes have been confused on the one hand 
with extensive and on the other hand with intensive 
magnitudes. As regards the former, the confusion is 
due to identifying the distensive magnitude of differ 
ence, say between the pitches C and G, with the stretch 
including all the intermediary pitches. This stretch 
illustrates what we have called an extensive whole ; 
and, in so far as it can be measured, its measure would 
be equivalent to that of the difference between C and 
G; i.e. its measure would be equivalent to that of a 
distensive magnitude, but the natures of the two are 
non-equivalent. As regards distensive and intensive 
magnitudes, these agree in so far as they both apply to 
qualities, and not obviously to things occupying a 



I 7 4 CHAPTER VII 

quantum of space or time or forming a linear or tem 
poral series; but it is necessary to distinguish them 
inasmuch as distensive magnitude requires the funda 
mental conception of different qualities which are yet 
comparable; while intensive magnitude requires what 
has sometimes been paradoxically described as the 
conception of a thing as merely qualitative, and yet as 
susceptible of quantitative variation. 

12. Having distinguished different kinds of mag 
nitude, we have now to consider how magnitudes of 
any given kind are to be compared; and we will begin 
by the simplest kind of magnitude, viz. that which can 
be predicated of a linear whole. 

Mr Russell deliberately adopts the view that the 
ultimate parts of a line are points, of which the number 
may be assumed to be 2 exp X, whatever be the magni 
tude of the line. In other words, any comparison of 
one line with another in regard to magnitude depends 
upon something other than the number of points which 
the lines contain. Hence the magnitude of an exten 
sive whole, as illustrated by a line, cannot be estimated 
in terms of pure or abstract number. In this respect 
it is of a totally different nature from a class, the mag 
nitude of which is entirely determined by the number 
of items it comprises, or by the number of exclusive 
sub-classes into which it may be divided. It follows 
then that magnitude, when applied to an extensive 
whole, has a different meaning from magnitude when 
applied to an extensional whole. For what I have 
called an extensive whole Mr Russell uses the term 
divisible whole, because the notion of dividing is 
essential to our conception of the relation of part to 



THE DIFFERENT KINDS OF MAGNITUDE 175 

whole, particularly in temporal and spatial applications. 
But, in discussing the principle required for comparing 
the magnitude of one line with that of another, he uses 
the phrase magnitude of divisibility. This phrase ap 
pears to me unfortunate inasmuch as it conveys no 
meaning : entities may be distinguished according as they 
do or do not possess the quality of divisibility ; and the 
term magnitude is of course required when we discuss 
whether one thing is greater or less than another. But 
I fail to see how we can regard one line as being greater 
than another on the ground that it possesses the quality 
of divisibility in a higher degree. It is quite certain 
that the number of parts into which a shorter line can 
be divided is exactly the same as the number of parts 
into which a longer line can be divided ; as also are 
the number of points in the one and in the other. The 
term magnitude of divisibility therefore appears to 
me merely to conceal what really is the problem in 
volved in comparing things having extensive magnitude ; 
namely the conception of equality of magnitude. 

13. What do we mean by the question, or how 
can we test, whether one given line or surface or bounded 
three-dimensional figure is greater or less than another ? 
Or again whether one stretch of hue or of pitch is 
equal to or greater than another ? In general, for two 
extensive wholes M and N of the same kind, if M in 
cludes but is not included in N it will be agreed that 
the magnitude of M is greater than that of N\ or 
briefly, the relation of superincident to subincident, 
whole to part proper, entails the relation of greater to 
less. But, if the wholes M and N are coexclusive, then 
no such test of equality or inequality can be directly 



176 CHAPTER VII 

applied ; and in order to compare their magnitudes in 
this case, we must be able to find parts of M that can 
be equated to one another as also to parts of N. This 
would provide us with a unit magnitude, in reference 
to which the magnitudes of M and N could be numeri 
cally compared. If we further assume that the wholes 
satisfy the strict criterion of continuity as defined by 
Cantor, then the series of numbers rational and irra 
tional will provide means for comparative measurement 
of all such magnitudes. On this assumption the only 
problem that remains is the provision of a test or 
definition of equality amongst unit parts. The possi 
bility of such a test must be separately examined for 
the three cases of spatial, temporal and qualitative 
stretches. As regards spatial wholes of one, two or 
three dimensions, the classical method is that of super 
position, the validity of which must be carefully con 
sidered. It is obviously absurd to think of the parts of 
space themselves as moving ; and hence the so-called 
method of superposition can only have practical signi 
ficance when we distinguish the material occupant of 
a place from the place which it occupies. When the 
material occupants of space are superposed one upon 
another, and the boundary of one is coincident with 
that of the other, they are said to be conterminous ; and 
when the boundary of one is subincident to that of the 
other, they may be said to be partially conterminous. 
Thus the outer boundary of a liquid and the inner 
boundary of a closed receptacle which it fills are coinci 
dent ; and, in this case, the volume occupied by the 
liquid is equal to the volume unoccupied by the receptacle. 
Again, if any two bodies have a common two-dimen- 



THE DIFFERENT KINDS OF MAGNITUDE 177 

sional boundary (which does not enclose a volume), then 
the boundary of the one has the same areal magnitude 
as that of the other ; and, if two bodies have a common 
one-dimensional boundary, then the boundary of the one 
has the same linear magnitude as that of the other. If, 
moreover, several different bodies can in either of these 
three ways be made conterminous with some one given 
body, the volume, area or length of the corresponding 
boundary of the one is equal to that of the other. But this 
predication of equality assumes that the volume of the 
receptacle, or of the areal or linear boundaries of the 
superposed bodies remains unchanged ; and the assump 
tion that in the course of time a material body does not 
change its spatial magnitude is in general invalid ; hence 
there is no literally logical justification for asserting 
equality or inequality in general, either with respect to 
the same body in different places, or with respect to the 
different places which the same body may occupy. 
Science in this case relies upon the constancy (under 
unchanged conditions) of the volume of certain bodies, 
and uses these as standards by which the changes of 
volume of other bodies are tested. In this process, we 
are continually acquiring more precise knowledge of 
causal conditions; but the final justification for com 
parisons of spatial magnitude is to be found in the 
coherency or consistency with which the systematisation 
of measurements and the construction of physical laws 
can be developed. The conclusion follows then that no 
directly logical test can be found, and we must be satis 
fied with the indirect principle according to which com 
prehensive universals are asserted on the mere ground 
that they do not lead to appreciable inconsistencies. 

J.L.II 12 



178 CHAPTER VII 

The problem of temporal magnitude, like that of 
spatial magnitude, is met first by the axiom that events 
that are conterminous at both ends have the same 
temporal magnitude, and secondly by the postulate that 
under identical causal conditions equal changes occupy 
equal lengths of time. We then employ some physical 
process, such as the movement of the hands of a watch, 
in which the mechanical conditions can be estimated 
with the closest approximation to exactitude, and adopt 
as standard time-units the times occupied by the changes 
thus effected. Conversely, where equal changes are 
effected during unequal times, we infer that the causal 
conditions are not identical. In all temporal changes, 
the means by which we can measure such changes 
as equal, itself depends upon the assumption that we 
can measure certain spatial, distensive or intensive 
magnitudes. 

Turning now to qualitative magnitudes, we have to 
consider by what method stretches of hue and pitch can 
be quantitatively compared. If we agree that the stretch 
from A to E is equal to that from C to G in a scale of 
pitches, this cannot be tested by any such method as 
that of superposition, for there is no distinction here 
corresponding to that between the place which is occu 
pied on the one hand, and that which is movable and 
can occupy indifferently one place or another on the 
other hand. If a qualitative stretch has magnitude, this 
involves the assumption that stretches of the same kind 
are comparable as greater or less. But how much 
greater, or by what ratio the two are to be compared 
must be determined, if at all, by some principle totally 
different from superposition. Mathematicians who have 



THE DIFFERENT KINDS OF MAGNITUDE 179 

written on this subject appear to agree in the view that 
two magnitudes may be comparable as greater or less, 
and yet not measurable in terms of number. But, if two 
stretches are mutually excludent, I can see no sense in 
which they can be compared as greater or less, unless 
we have a test of equality; and, when such test is 
forthcoming, a numerical measurement seems to me 
immediately to follow. Numerical measurement is 
not a merely arbitrary one-one correlation between 
numbers and magnitudes : for such correlation could 
only mean that for the greater magnitudes we apply 
higher numbers, and the precise numbers which we 
correlate would be absolutely arbitrary. Hence it 
appears to me that if a specific one-one numerical 
correlation has an objective ground, according to 
which it is to be preferred to any other, this must 
be because we have adopted some principle for de 
termining a correct quantitative unit For example, 
if we prefer the absolute measurement of tempe 
rature to the thermometric measurement as deter 
mined say by the changes of volume of mercury, this 
is because we believe that the differences of tempe 
rature indicated by the former scale do correspond 
to really equal differences of magnitude, whereas the 
other does not. Readers of Clerk Maxwell s Heat 
will learn that the absolute measurement of tempera 
ture depends upon measurements of heat and work, 
which are complex quantities, being partly extensive 
and partly intensive. In all such cases, where we can 
not directly measure a cause or an effect, we measure 
it indirectly in terms of its effect or cause (as the case 
may be). 



12 2 



i8o CHAPTER VII 

14. The entities of which either extensive, disten- 
sive or intensive magnitude can be predicated alone 
may be termed simple or simplex, and from these kinds 
of entity we now pass to those which may be called 
compound or complex, on the ground that two or more 
magnitudes are combined in our conception of the 
quantity of the resultant complex. These latter may be 
illustrated by light sensations, which vary intensively 
according to their brightness, distensively as regards 
their hue, and in yet a third respect according to the 
proportion in which the chromatic and achromatic factors 
are combined to produce different degrees of saturation. 
Similarly sound sensations vary intensively according 
to their loudness, distensively as regards their pitch, 
and as regards timbre or klang-tint in accordance with 
the proportional intensities of their constituent tones, 
under-tones and over- tones. It is convenient to speak, 
then, of light and sound sensations as three-dimensional, 
in the sense that there are three distinct determinables 
under which any such sensation can be defined and 
quantitatively estimated. But the simplest case of a 
three-dimensional quantity is space. In space we may 
take three arbitrary directions ; and, according to the 
ordinary view, the magnitudes (i.e. lengths) along these 
directions have the unique characteristic of being com 
parable. Any point in a space of three dimensions is 
therefore assignable by three ordinates drawn in deter 
mined directions from a given point as origin. In this 
way a surface in three dimensions, or a line in two 
dimensions, differs from what is called a graph, in that 
the magnitudes represented by the ordinates of a point 
in the graph are of different kinds and therefore incom- 



THE DIFFERENT KINDS OF MAGNITUDE 181 

parable. For example: a graph representing the co 
variation of work done and hours expended uses two 
incomparable magnitudes. 

The general topic which we have now to consider 
is that of a derived quantity that is constructed by some 
kind of combination of other quantities. I n constructing 
a quantity comparable with each of those combined, the 
processes of addition and subtraction can alone be 
applied; and, conversely, addition and subtraction can 
only be applied to comparable quantities. Such addition 
and subtraction may be termed concrete, in antithesis 
to abstract in which pure numbers are concerned whose 
sum or difference is also a pure number. Now I shall 
maintain that processes analogous to multiplication and 
division may be employed in constructing a quantity of 
a different kind from any of those that are combined in 
its construction ; and such multiplication or division 
may also be called concrete. Thus, considering first the 
three notions of length, area, and volume, I shall say 
that the multiplication of two differently directed lengths 
constitutes an area, and that of three differently directed 
lengths constitutes a volume. Here we are extending 
the operation called multiplication beyond its primary 
use. For, while it is universally agreed that we may 
multiply a pure number by a pure number, in con 
structing another pure number, or a quantity of any 
kind by a pure number, in constructing a quantity of 
the same kind, yet most mathematicians have refused 
to allow that by multiplying one quantity by another 
we may construct a third quantity different in kind from 
both the quantities multiplied. They maintain that what 
is multiplied is the numerical measure of the quantities 



182 CHAPTER VII 

and not the quantities themselves. Similarly with regard 
to division: it is agreed that we may divide a pure 
number by a pure number in constructing another pure 
number, or a quantity of any kind by a pure number in 
constructing a quantity of the same kind; but we are 
prohibited from dividing one quantity by another in 
constructing a quantity different in kind from the quan 
tities divided. In this case too, the so-called division 
is regarded as a division not of the quantities but of 
their numerical measures. My first objection to this 
view is that it offers no means of distinguishing between 
the multiplication or division of a quantity by a pure 
number, which yields a quantity of the same kind, from 
that very different kind of multiplication or division 
which yields a quantity different in kind from those 
multiplied or divided. My disagreement, however, with 
the almost unanimous opinion of mathematicians may 
perhaps be considered merely verbal ; but the view that 
I maintain is, I think, based upon an important logical 
principle. Apart from any conception of numerical 
measurement which adopts numbers, integral, rational 
and irrational, it appears to me that we must conceive 
the process of multiplying say a foot by an inch (which 
involves no idea of number) as a construction by which, 
from two magnitudes of the same kind, a third magni 
tude of a different kind is derived. If no such magni 
tude were presented to perception or thought, it would 
follow that no meaning could be attached to such multi 
plication; but, inasmuch as an area is a genuine object 
of thought construction, I see no insurmountable objec 
tion to speaking of the process of multiplication as that 
by which area, for instance, is constructed out of two 



THE DIFFERENT KINDS OF MAGNITUDE 183 

directed lengths, or volume out of three. The mathe 
maticians who reject this idea hold that the notion 
of units of different kinds is sufficient, without intro 
ducing the multiplication or division of units. It is 
agreed that the area of a rectangle whose sides are of 
unit length is a unit area, and the volume of a cube 
whose sides are of unit length is a unit volume. In this 
way the numerical measures of area and volume are ob 
tained by multiplying the numerical measures of their 
sides; but in my view we must allow that the lengths 
themselves are multiplied, for otherwise we could not 
distinguish the different kinds of magnitudes constructed, 
since where only abstract numbers are concerned only 
abstract numbers are constructed, and there is nothing 
to indicate the difference between one quantity thus 
derived and another. Passing from concrete multipli 
cation to concrete division we have what may be thought 
a more interesting and certainly a wider application of 
the same general principle. Those quantities which are 
derived by dividing one kind of quantity by another 
may be called rate-quantities, or in certain cases degree- 
quantities. A rate-quantity is expressed in familiar 
English by the Latin word per, of which it is easy to 
multiply examples: e.g. space traversed per second, 
wages earned per hour, pleasure experienced per minute, 
pressure per square foot, mass per cubic foot. The two 
constituent quantities in this kind of division may be 
themselves complex or of different kinds, extensive, 
distensive or intensive ; but so far as the conception of 
concrete division is concerned, no logical distinctions 
are required in analysing the general notion of a rate- 
quantity. Each of the rate-quantities constructed by 



i&4 CHAPTER VII 

this species of division is a quantity of a different kind 
from the quantities of which it is constituted; and, as in 
multiplication, it is always useful for arithmetical pur 
poses to adopt as the derived unit-quantity that which 
is constructed out of fundamental unit-quantities. The 
general term rate which I have introduced is in common 
use: thus we mean by the rate of wages the quantum 
of wages earned per unit of time, by the rate of speed 
the quantum of length traversed per unit of time, and 
by the rate at which pleasure is being experienced, the 
quantum of pleasure per unit of time these being cases 
in which the rate is estimated in reference to time. 
Again the rate called hydrostatic pressure is the quantum 
of pressure per unit of area; the rate called density is 
the mass per unit of volume. The term degree, which 
is sometimes used instead of rate, is ambiguous inas 
much as it is often used as equivalent to intensity ; but 
the terms rate and intensity or degree ought to be 
clearly distinguished, because the notion of intensity 
refers to a single determinate quality, whereas rate is 
always constituted out of two distinguishable quantities ; 
moreover the notion of rate, which involves concrete 
division, is always correlated with concrete multiplica 
tion. For example velocity, i.e. rate of movement, which 
involves the division of space by time, involves the 
converse process of multiplying velocity by time in 
constructing space. But if we conceive of velocity 
only by its numerical measure, confusion results be 
tween an abstract number on the one hand and the 
very many different kinds of quantity on the other 
hand that may be measured by the same abstract 
number. 



THE DIFFERENT KINDS OF MAGNITUDE 185 

15. The practical importance of recognising con 
crete multiplication and division is best indicated by 
explaining what is meant by the algebraical dimensions 
of a quantity. We have already spoken of dimensions 
in its geometrical sense; thus an area is of dimension 
two in regard to length, a volume of dimension three in 
regard to length. Symbolising the dimension length by 
[Z] that of area is symbolised by [Z 2 ] and that of volume 
by [Z 3 ]. Similarly velocity, i.e. length per time, is di- 

mensionally ~. or [Z] . [T^ 1 ]; acceleration, i.e. velocity 



per time is [Z]. [7"~ 2 ]; density, i.e. mass per volume is 



T, i- e - D^]-D~ 8 ]; momentum, i.e. mass x velocity 



is [^/][Z] [7^~ 2 ]; force, i.e. mass x acceleration is 
[J/]. [Z]. [7^" 2 J; hydrostatic pressure, i.e. force per 
area is \_M~\ . [Z" 1 ] [7"~ 2 ], etc., etc. Now the one rule as 
regards dimensions is that the additions and subtractions 
that are involved in a quantitative equation must always 
operate upon homogeneous quantities; i.e. upon quan 
tities all of which have the same dimensions these 
dimensions being generally expressed in terms of the 
three fundamental incomparables mass, length, and time. 
Regarding multiplication and division, in accordance 
with my view, as real operations performed upon con 
crete quantities, the square bracket in the above symbols 
stands for a concrete unit. For example the velocity 

320 ft. i6ft. 16 r 

320 feet per 60 seconds means - = -- = ot 

60 sec. 3 sec. 3 

unit velocity. Those mathematicians who hold that 
such an expression as ft. -r sec. is meaningless have to 
maintain that the mathematical equations which are 



186 CHAPTER VII 

used to express physical facts are concerned only with 
the numerical measurement of concrete quantities, 
whereas I hold that they are concerned with the concrete 
quantities themselves. 

1 6. There is one very unique case in concrete 
division, viz. where the dividend and divisor are quan 
tities of the same kind. In general the result of such 
division is to construct a pure ratio, i.e. a magnitude 
which, when entering as multiplier or divisor of a quan 
tity of any kind yields a quantity of the same kind, like 
the processes of addition and subtraction of quantities. 
But when a length is divided by a length, or an area by 
an area, we often intend the result of such division to 
represent an angle. It is therefore necessary to dis 
tinguish those cases in which the division of a length 
by a length represents a mere ratio, from those in 
which it represents an angle. In the former case, the 
quotient being a pure number can be used as a multi 
plier or divisor for a quantity of any kind whatever; 
but in the latter case this is never possible ; one angle 
can only be mathematically combined with another 
angle, and this only by the operation of addition or of 
subtraction. The further complication in respect of the 
measurement of an angle is that this measurement may 
be used in different algebraical applications alternatively 
either as an abstract ratio or as a concrete quantity, which 
is denoted by the term angle. But the special question 
which, in my view, requires a clear answer is how to 
distinguish the process of dividing length by length that 
yields a mere ratio, from what appears to be the same 
process and yet yields an angle. The answer seems to 
be that when we are merely comparing two lengths 



THE DIFFERENT KINDS OF MAGNITUDE 187 

which may be said to be dissociated, their comparison 
yields a mere ratio, while when connecting two asso 
ciated lengths in the process of division, we are con 
structing an angle. Thus, when we define the magni 
tude of an angle by the ratio of the arc of a circle to its 
radius, the arc and the radius are associated in our con 
ception of the mode in which the angle is constructed; 
but when we are merely comparing the length of one 
line with that of any other, no natural association between 
the two lines is involved. The same holds of the dif 
ferential coefficient dy by dx, when used in geometry 
to represent the slope of a tangent of a curve, which is 
a concrete quantity in the same sense as the quotient 
foot by second representing velocity. 

17. To sum up: Of the different kinds of magni 
tude, the first division is between abstract and concrete, 
abstract magnitudes being represented by pure numbers, 
these falling into the three divisions of integral, rational 
and irrational. Amongst concrete quantities namely 
those that involve conceptions obtained from special 
kinds of experience, and which are therefore not purely 
logical we distinguish the fundamental or primitive 
from the complex or derivative ; the former being sub 
divided into extensive, distensive and intensive magni 
tudes, out of which the various derived or complex 
quantities have been shown to be constructed by opera 
tions analogous to arithmetical multiplication and divi 
sion. These complex magnitudes fall again into different 
kinds, the distinctions between which may be always 
indicated by expressing the quantity dimensionally, i.e. 
as involving a concrete product of different fundamental 
quantities, each entering with a positive or negative 



i88 CHAPTER VII 

index. Finally a fundamental distinction has been drawn 
between addition or subtraction on the one hand and 
multiplication or division on the other; inasmuch as the 
quantities added or subtracted must be of the same kind, 
i.e. represented as dimensionally equivalent; whereas 
the operations of multiplication and division yield a 
quantity different in nature from its factors, which, 
however, together determine its nature. Throughout 
the whole discussion of concrete magnitudes, the diffi 
cult problem of defining or testing equality has been 
examined for each fundamentally distinct kind of quan 
tity. The treatment has been comparatively elementary, 
the reader being referred for more subtle distinctions 
and analyses to works which deal primarily with mathe 
matics and its philosophy. 



CHAPTER VIII 

INTUITIVE INDUCTION 

i. INDUCTION in general may be contrasted with 
deduction in that for a universal conclusion deduction 
needs universal premisses, whereas in induction a uni 
versal conclusion is drawn from instances of which it is 
a generalisation. Here the emphasis is upon the word 
instances, because although the customary account of 
deduction is that the range of the conclusion is identical 
with that of the narrowest of the premisses, yet de 
duction must include cases in which the range of the 
conclusion is not identical with that of any one of the 
premisses, and may even be wider than the widest of 
them. Actually the antithesis between inductive and 
deductive inference is not so fundamental as that between 
demonstrative and problematic inference ; for every 
form of induction, except the problematic, is based upon 
the same fundamental principles (and these alone), as 
syllogism and other forms of deduction ; whereas it is 
impossible to establish a theory of problematic induc 
tion, without recourse to certain postulates that are not 
involved in either form of demonstration, whether de 
ductive or inductive. Now the fundamental principles 
which underlie demonstrative forms both of induction 
and deduction are themselves based upon a kind 
of inference which may be called intuitive induction. 
This process is not limited to the establishment of the 
principles of demonstration, but applies also to certain 
material as well as formal generalisations. 



CHAPTER VIII 

We have so far referred to two types of induction, 
viz., intuitive and demonstrative ; it will be convenient 
to distinguish in all four varieties, namely intuitive, 
summary, demonstrative and problematic. Of these 
the three former will be discussed in the present Part 
of this work, but problematic induction will be examined 
in detail in a separate Part, on the ground, specified 
above, of its dependence upon special postulates. 

2. Before treating the main topic of this chapter, 
we must discuss the necessarily preliminary process 
known as abstraction, the nature of which was a special 
subject of philosophical and psychological controversy 
amongst James Mill and his contemporaries. The dis 
cussions of that date started from the supposition that 
what was presented in our earliest acts of perception 
was a combination of impressions from different senses, 
such as those of sight and touch. From this pre 
supposition, upon which both parties were agreed, the 
difficulty was raised as to how the percipient could 
single out an occurrent impression of one sense from 
the concurrent impressions of other senses. This pre 
supposition, however, is fundamentally mistaken. For, 
in fact, our earliest acts of attention, which yield any 
product that could be called a percept, are directed to 
impressions of one sense at one time, and to impressions 
of another sense at another time. For example, the 
child when interested in the colour of a ball, is attending 
to his visual impressions apart from any motor or tactual 
sensations that he may be experiencing in handling 
the ball ; that is to say, his attention is from the first 
exclusive, and it is only in further progress of attentive 
power that his attention becomes inclusive. The atten- 



INTUITIVE INDUCTION 191 

tion that includes visual with tactual impressions is a 
higher and later process than the attention which is 
directed either exclusively to the visual impressions or 
exclusively to the tactual impressions. The fact that 
we can and do attend to impressions of one order in 
disregard of concurrent impressions of other orders, 
explains how our primitive perceptual judgments, from 
the first, assume a logically universal form. For, in 
predicating a determinate colour, for instance, of any 
given impression, there is a recognition that the same 
determinate can be predicated of all impressions which 
agree with the given impression in respect of colour, 
however much they may disagree in other respects. 
Now, if this be granted, it has an important bearing 
upon another serious historical controversy namely 
that between Mill and his opponents as to the founda 
tions of geometry. Both parties to this dispute started 
with an obscure view, that there was an opposition 
between intuition and experience ; whereas in truth 
intuition is a form of knowledge, in relation to which 
experience is the matter. The intuitionists seem to 
have held that the intuitive form of knowledge involved 
no reference to experience ; whereas the empiricists 
forgot, when relying upon experience as the sole factor 
in knowledge, that knowing is a mode of activity, and 
therefore not of the same nature as sense-experience 
which is merely passive or recipient. The truth is that 
when we have asserted a predicate of a particular, we 
have apprehended the universal in the particular, in the 
sense that the adjective is universal and the object of 
which it is predicated is particular. 

3. There is another sense in which we may be 



I 9 2 CHAPTER VIII 

said directly to apprehend the universal in the par 
ticular, namely in regard to certain classes of proposi 
tions, where the terms universal and particular apply 
to the propositions themselves, and not to the distinc 
tion between the subject and the predicate within the 
proposition. It is at this stage that we pass, in our 
discussion, from abstraction to our main topic, viz., 
abstractive or intuitive induction. The term intuitive 
is taken to imply felt certainty on the part of the 
thinker ; and it is characteristic of propositions estab 
lished by means of intuitive induction that an accumu 
lation of instances does not affect the rational certainty 
of such intuitive generalisations. The procedure by 
which these generalisations are established may be 
shown by psychological analysis to involve an inter 
mediate step by which we pass from one instance to 
others of the same form and in this passage realise 
that what is true of the one instance will be true of all 
instances of that form. 

4. Two types of intuitive induction may be dis 
tinguished, experiential and formal, although these types 
are not precisely exclusive of one another. 

The experiential type of intuitive induction may be 
illustrated from our immediate judgments upon sense- 
impressions and the relations amongst them. For 
example, in judging upon a single instance of the 
impressions red, orange and yellow, that the qualitative 
difference between red and yellow is greater than that 
between red and orange (where abstraction from shape 
and size is already presupposed) this single instantial 
judgment is implicitly universal ; in that what holds of 
the relation amongst red, orange and yellow for this 



INTUITIVE INDUCTION 193 

single case, is seen to hold for all possible presenta 
tions of red, orange and yellow. Again in immediately 
judging that a single presented object, whose shape is 
perceived to be equilateral and triangular, is also equi 
angular (where abstraction from colour and size is pre 
supposed) we are implicitly judging that all equilateral 
triangles are equiangular. Similarly when judging for 
a single instance that the sounds A, C, F, produced, 
say, from the human voice, are in an ascending scale 
of pitch, we are implicitly judging that all sounds 
apart from differences of timbre or loudness such as 
those produced by the violin or piano that can be 
recognised as of the same pitches A, C, F, are also in 
an ascending order of pitch. The universality of these 
experiential judgments extends over imagery as well 
as sense impressions : the fact that we can identify a 
specific image as corresponding to a specific impression 
is sufficient to enable us directly to transfer our judg 
ments about the relations amongst impressions to those 
amongst the corresponding images. These elementary 
illustrations show that intuited universals about colours 
and pitches are of the same epistemological nature as 
those about geometrical figures, in that the judgment 
upon a single presented instance is sufficient for the 
establishment of a universal extending in range over 
imagery as well as impression. 

5. Passing now to other experiential judgments, 
which are not merely sensational, we may illustrate 
intuitive induction from introspective judgments. 
For instance, when I judge that it is the pleasure of 
this or that experience which causes me to desire it, 
I am implicitly universalising and maintaining that the 

j. L. ii 13 



I 9 4 CHAPTER VIII 

pleasure of any experience would cause me to desire 
it. And again, when I judge that the greater resultant 
desire for one possible alternative than for any other 
causes me to will that alternative, I am judging that 
this will hold for all my volitional experiences. An 
important sub-class of experiential judgments which 
are intuitively inductive consists of moral judgments. 
Thus, when anyone judges that a certain act charac 
terised with a sufficient degree of precision is cowardly, 
or dishonest, or generous, he is implicitly judging that 
all acts of the same specific character would be charac- 
terisable by the corresponding moral attribute. That 
this is not a case of mere abstraction is clear when we 
consider that the characteristics used to define the 
nature of the action are other than ethical, and that 
the judgment is therefore synthetic. This intuitive 
aspect of moral judgments assumes importance as re 
conciling the two forms of ethical intuitionism to which 
Sidgwick refers as Perceptual and Dogmatic, the first 
of which stands for the particular, and the second for 
the universal, intuition. For, in my view, the Dogmatic 
form of intuition is not genuinely intuitive except so 
far as it is based on the Perceptual. Instead, therefore, 
of distinguishing moralists according to what they hold 
to be the nature of an ethical intuition, it is more im 
portant to distinguish them according as they base 
their doctrine upon genuinely intuitive judgments, e.g. 
Kant ; or upon judgments accepted on authority as 
expressions of the voice of God, e.g. Butler. 

6. The gulf between experiential and formal 
intuition is bridged by considering certain intermediary 
forms of intuitive apprehension in which, according as 



INTUITIVE INDUCTION 195 

the range of universality increases, we depart further 
from the merely experiential and approach nearer to 
the merely formal type. A typical case is the merely 
experiential judgment that red and green cannot both 
be predicated of the same visual area by one person 
at one time. The judgment is first universalised when 
the experient sees that the same holds of all cases of 
the specific determinates red and green. But this judg 
ment almost immediately passes into the wider universal 
that any two different determinates under the deter- 
minable colour are similarly incompatible. And when 
lastly the experient extends the range of his judgment 
to all determinables, he has reached a formal intuition, 
namely that any two different determinates under any 
determinable are incompatible. 

To this formal type of intuition belong all intuitively 
apprehended mathematical, as well as purely logical, 
formulae. For instance, the algebraical formula known 
as the Distributive Law is intuitively reached in some 
such way as this : perceiving that 

3 times 2 ft. + 3 times 5 ft. = 3 times (2 ft. + 5 ft.) 
we immediately realise that 

4 times 7 days + 4 times 9 days = 4 times ( 7 days + 9 days), 
and in this step we are virtually apprehending the Dis 
tributive Law symbolically expressed thus : 

n times P + n times Q = n times (P+Q) 
where n stands for any number, and P and Q for any 
two homogeneous quantities. 

A logical example of a similar nature is the formula 
of the simple conversion of particular affirmative pro 
positions. This is reached by perceiving, for instance, 

132 



196 CHAPTER VIII 

that Some Mongols are Europeans would imply that 
Some Europeans are Mongols, and at the same time 
that 

Some beings incapable of speech have the same 
degree of intelligence as men would imply that Some 
beings having the same degree of intelligence as men 
are incapable of speech. 

This leads to the virtual apprehension of the universally 
expressed implication: 

Some things that are p are q would imply that 
Some things that are q are/ 

where/ and q stand for any adjective. 

7. This example of the establishment of logical 
formulae by means of intuitive induction has an educa 
tional importance in correcting a certain prevalent con 
ception of the function of logic. What is called formal 
or deductive logic is usually taught by first presenting 
general principles in a more or less dogmatic form, with 
the result that the learner is apt to use these principles 
merely as rules to be applied mechanically in testing 
the validity of logical processes. Instead of leading him 
to conceive of these rules as externally imposed impera 
tives, an appeal should be made to him to justify all 
fundamental principles by the exercise of his own 
reasoning powers ; and this exercise of power will in 
volve the process of intuitive induction. 



CHAPTER IX 

SUMMARY (INCLUDING GEOMETRICAL) INDUCTION 

i. THE term summary induction is here chosen 
in preference to what, in the phraseology of the old 
logicians, was called perfect induction, to denote a 
process which Mill regarded as not properly to be called 
induction ; on the ground that the conclusion does not 
apply to any instances beyond those constituting the 
premiss. Mill s contention can certainly be justified in 
asmuch as the process involves precisely the same 
logical principles, and these alone, that govern ordinary 
deduction. In fact, the process of summary induction 
may be expressed in the form of a syllogism in the first 
figure. For example : 

Major Premiss. Sense and Sensibility and Pride 
and Prejudice and Northanger Abbey and Mansfield 
Park and Emma and Persuasion deal with the 
English upper middle classes. 

Minor Premiss. Every novel of Jane Austen is 
identical either with Sense and Sensibility or with 
Pride and Prej udice or with Northanger Abbey or with 
Mansfield Park or with Emma or with Persuasion. 

Conclusion. . . Every novel of Jane Austen deals 
with the English upper middle classes. 

Here the enumeration standing as subject in the major 
premiss is the same as the enumeration standing as 
predicate in the minor premiss. But, in the former, 
reference is made to every one of the collection, in the 



198 CHAPTER IX 

latter to some one or other. This precisely corresponds 
to the characteristic of first figure syllogism; namely 
that the middle term is distributed as subject of the 
major and undistributed as predicate of the minor. In 
text-book illustrations of perfect induction the minor 
premiss is almost invariably omitted, because the illus 
trations chosen such as the Apostles or the months of 
the year are so familiar that the completeness of the 
enumeration is assumed to be known by every ordinary 
reader and therefore does not require to be expressed 
in a separate minor premiss. The same process is 
exhibited by an example in which each of the items 
enumerated is a universal instead of being a singular : 

Every parabola and every ellipse and every hyper 
bola meet a straight line in less than 3 points. 

Every conic section is either a parabola or an ellipse 
or a hyperbola. 

.*. Every conic section meets a straight line in less 
than 3 points. 

2. Another case of perfect induction, which has 
specific bearing upon induction in general, may be 
expressed symbolically in the following syllogism: 

s 1 and ^ 2 ... and s n are/. 

Every examined case of m is identical either with 

s 1 or with s 2 ... or with s n . 
. . Every examined case of m is/. 

A summary or perfect induction of this form is the 
necessary preparatory stage in gathering together the 
relevant instances for establishing an unlimited generali 
sation. For the conclusion thus obtained, constitutes the 
premiss from which we directly infer, with a higher or 
lower degree of probability, that Every case of m is// 



SUMMARY INDUCTION 199 

Whewell pointed out the importance and difficulty of 
discovering the concept p under which the instances 
are colligated. He, in agreement with other critics of 
Mill, accordingly held that the process of induction was 
completed in the discovery of this colligating concept, 
on the ground that this process alone required some 
thing like genius to perform, while it is the easiest 
thing in the world to pass from every examined instance 
to every instance. Mill, on the other hand, considered 
that this process only supplied the requisite premiss for 
a genuine inductive inference. To illustrate his view, 
Whewell had chosen Kepler s famous discovery of the 
formula for the orbit of the planets, and it was towards 
this illustration that Mill directed his criticism. Ex 
pressed in terms of the above used symbols. 

Let m stand for positions of a certain moving planet/ 
s lt s 2 ... s n ,, ,, the several observed positions/ 
and / ,, ,, being a point on a certain ellipse. 
The syllogism which expresses the process of perfect 
induction used by Kepler will then be as follows: 

Each of several observed positions is a point on 
a certain ellipse. 

Every examined position of a certain moving planet 
is identical either with one or with another of these 
several observed positions. 

. . Every examined position of the moving planet 
is a point on that ellipse. 

This formula had not been discovered by any previous 
astronomer, and, on the grounds already assigned, 
Whewell maintained that the discovery constituted the 
completion of the induction. To this Mill demurred, 
because by induction he meant a process in which the 



200 CHAPTER IX 

conclusion is an unlimited universal extending beyond 
examined instances ; he, however, failed to observe that 
Kepler had actually gone beyond the examined instances 
and had described the complete orbit of the planet by 
inferring that what held of the examined positions 
would hold of all the interpolated positions. Kepler 
had thus unconsciously made a genuine induction in the 
sense required by Mill. Whewell was concerned with 
the art of discovery, and therefore held that the essen 
tial factor in induction was the discovery of the colli 
gating concept; whereas Mill was concerned with the 
science of proof, and therefore held that the essential 
factor in any induction (that was not merely formal or 
demonstrative) was the inferential extension from ex 
amined to unexamined instances. 

3. Having illustrated the process of summary (or 
perfect) induction by familiar examples, in which the 
conclusion applies to a finite number of cases which are 
enumerable, we proceed to consider a more interesting 
type of summary induction in which the conclusion 
applies to an infinite number of cases which are non- 
enumerable. This type occurs in geometrical proofs of 
geometrical theorems, and has been more or less con 
fused on the one hand with merely intuitive, and on the 
other hand with problematic induction. It differs, how 
ever, from the former in that its conclusion cannot be 
reached from an examination of one or of a few instances; 
and from the latter in that the conclusion does not 
extend beyond the range of the examined instances 
these being apprehended in their infinite totality. 

It is well known that there are two modes by which 
geometrical theorems may be proved, viz. analytical 



SUMMARY INDUCTION 201 

and geometrical. Strict analytical proof has the same 
logical character as algebraical proof, and comes under 
the head of functional deduction. Such proofs do not 
require the aid of geometrical figures. But the geo 
metrical method of proof depends essentially upon the 
use of such figures. It may further be pointed out that 
the analytical method has an indefinitely wider scope 
than the geometrical. For example, by employing mere 
analysis we can construct spaces of various different 
forms other than Euclidian; and certainly a geometrical 
method would be impossible except as applied to our 
space which is presumed to be Euclidian. The actual 
procedure in constructing any non-Euclidian space is to 
bring forward some four or five axioms which must be 
(a) independent of one another, and (K) mutually con 
sistent. These axioms, however, are not put forward 
categorically, but purely hypothetically ; it follows, there 
fore, that the theorems which, for convenience are said 
to be deduced from the axioms, should be more strictly 
said to be implied by the axioms. Such systems, there 
fore, are throughout implicative and not inferential. In 
other words, a supposed space, definable by any chosen 
set of axioms, would have such and such other charac 
teristics which these axioms would formally imply. On 
the other hand, the geometrical method is a method of 
proof or inference, inasmuch as we accept its conclusions 
as true only because we have accepted its axioms as 
true. 

4. We must therefore examine the process by 
which the axioms of geometry are established. These, 
qua axioms, are not reached by deduction ; and, since 
they are universal in the specific sense that they apply 



202 CHAPTER IX 

for an infinite number of possible instances, it would 
seem that some form of induction is required for their 
establishment ; unless we adopt the view that they are 
obtained by a process which embraces all possible cases 
in a single act of direct intuition. This latter appears 
to be the view of Kant who held, as regards geometry, 
that our intuitions are from the first universal, and that 
they therefore function as premisses for deducing any, 
or any other, given case. 

In order to examine this question let us take the 
familiar axiom conveniently expressed in the form: 
Two straight lines terminating at the same point cannot 
intersect at any other point. This is the most important 
axiom which does not hold of non-Euclidian spaces in 
general. Independently, however, of the nature of any 
other kind of space, the axiom certainly represents the 
manner in which we actually intutte ozcr space, whether 
falsely or truly. Now this axiom, in its universality, 
can be established only by means of imagery and not 
by mere perception; for the compass over which the 
axiom holds is beyond the range of actual perception. 
For in the first place it is only through imagery that we 
can represent a line starting from a certain point and 
extending indefinitely in a certain direction ; and, in the 
second place, we cannot represent in perception the 
infinite number of different inclinations or angles that 
a revolving straight line may make with a given fixed 
straight line. We may, however, by a rapid act of 
ocular movement represent a line revolving through 
360 from any one direction to which it returns. In this 
imaginative representation the entire range of variation, 
covering an infinite number of values, can be exhaus- 



SUMMARY INDUCTION 203 

tively visualised because of the continuity that charac 
terises the movement. It is only if such a process of 
imagery is possible that we can say that the axiom in 
its universality presents to us a self-evident truth. It 
is therefore this species of summary induction that is 
employed to establish geometrical axioms differing, as 
explained above, on the one hand from mere intuitive 
induction, inasmuch as one or a few specific cases would 
not constitute an adequate premiss ; and, on the other 
hand, from induction in Mill s specific sense, since the 
conclusion does not go beyond the premisses taken in 
their totality. 

5. I shall further maintain that if, in the course of 
a geometrical proof which may involve several succes 
sive steps, the perception or image of a figure is required 
for any single step, this is because we have to go 
through precisely the same process of summary induc 
tion, embracing an infinite number of specialised cases 
of which the figure under inspection is one all of these 
being included in the subject of the universal conclusion 
to be proved at that step. Speaking generally, in any 
one demonstrative step, the major premiss is a universal 
previously established, and from this universal major it 
is required to establish a new universal conclusion. It 
is obvious that this can only be done by means of a 
universal minor; and it is in the establishment of the 
universality of the minor that consists the logical func 
tion of the figure. The arbitrarily chosen figure under 
inspection can only be used as a minor term to prove 
the conclusion about that single figure; and hence, to 
obtain the required universal conclusion, the minor 
must be universalised by the same logical process that 



204 CHAPTER IX 

is used for establishing the explicit axioms. Now the 
Euclidian geometry might have been established by 
purely analytical methods ; provided first, that a sufficient 
number of axioms had been explicitly formulated ; and 
secondly that each of these axioms had been established 
for itself by the process of summary induction. Such 
an analytical system would dispense with the use of 
figures as objects either of perception or of imagery in 
the course of the proof , these being only required in the 
process of establishing the axioms themselves. 

To show by specific illustration how the geometrical 
proof uses a figure, we will select a very frequently 
assumed, but not explicitly stated axiom, which, in 
Euclid s proofs is required to supplement the explicit 
axiom the whole is greater than its part, or more 
precisely, the whole is equal to the sum of its parts. 
Before this explicit axiom can be used, we must be 
satisfied that the two elements of the figure, one of 
which is to be greater than the other, do stand in the 
relation of whole to part. The axiom to which I refer 
is actually employed by Euclid and most geometricians 
in the propositions numbered 5, 6, 7, 16, 18, 20, 21, 24, 
and 26 in Euclid, Book I. It may be formulated as 

follows : The angle sub 
tended at any point by a 
part of a line is part of the 
angle subtended by the 
whole line. If the reader 
is not familiar with this 
new axiom, he must go 

through a process in which he imagines a line revolving 
in a plane through a point (O) from some initial direc- 




SUMMARY INDUCTION 205 

tion (OA) to a final direction (OC), so that it will 
intersect the whole line (AC] in a series of successive 
points. In this way, and in this way alone, can he 
accept the universality of the required conclusion that 
the angle AOC is greater than the angle A OB. In 
Euclid s theorems enumerated above it will be found 
that this axiom is required in every case to establish 
the conclusion that a certain angle is greater than 
another ; and that this conclusion is a necessary step in 
the further progress of each proof. 

Geometrical induction involves, in addition to the 
summary process above explained, two further pro 
cesses which are of the nature of intuitive induction, 
as explained in the preceding chapter. Of these two, 
the first is concerned with absolute position, the second 
with absolute magnitude. Thus, having reached a 
universal by summary induction limited to figures oc 
cupying a certain position, it is by intuitive induction 
that we pass to figures of the same specific shape and 
magnitude occupying any other possible position ; and 
again from a figure imaged as having a certain magni 
tude, to figures of the same specific shape but of any 
other possible magnitude. I have described these two 
processes as of the nature of intuitive induction, in 
which we universalise by abstracting from variable 
position and from variable magnitude ; but they might 
otherwise be regarded as involving the conception of 
position and magnitude as being not absolute but 
relative to the percipient s own position and to his dis 
tance from the figure depicted in imagination. 

6. Having illustrated the proper use of the geo 
metrical figure, we shall proceed to illustrate what may 



206 CHAPTER IX 

be called its abuse ; and give, by means of a figure, an 
alleged proof that every triangle is isosceles : 

To prove that every Triangle is Isosceles. 

Let the bisector of 
the vertical angle A 
meet the perpendicular 
bisector of the base^C, 
whose middle point is 
D, at the point O. Join 
BO, CO, and draw OE 
perpendicular to AC, 
B D C and OF perpendicular 

toAJB. Then, 

(i) the triangles BOD, COD, are congruent; for 

OD is common; and 
LBDO= LCDO-, 




(2) the triangles AOE, A OF, are congruent; for 
AO is common; L OAE = L OAF-, and 
LOEA = LOFA-, 



(3) the triangles COE, BOF are congruent; for, 
by (i) CO = BO-, and by (2) OE=OF- and 
hence CO 2 -OE* = BO* - OF* ; 

i.e. (since CEO and BFO are rt L ) CE* = BF*. 

Hence, by (2) AE = AF*nd 9 by (3), CE = BF\ 
. . by addition AC=AB. Q.E.D. 

Here we see that the axiom : the whole is greater 
than its part is used in its more precise form, the 
whole is equal to the sum of its parts. Now before we 
can state as regards the straight line AFB, that 



we must be sure that AF, FB are really parts of AB\ 



SUMMARY INDUCTION 



207 



whereas if F was beyond AB, then AF would be the 
whole and AB, BF would be its parts. 

The fallacy incurred in this proof arises from the 
mistaken intuition that the bisector of the vertical angle 
A meets the perpendicular bisector of the base BC at 
a point O inside the triangle. By drawing an incorrect 
figure and thus convincing ourselves of the false con 
clusion, we had unconsciously universalised from the 
figure before us that for every case the two bisectors 
would meet at a point within the triangle, this being in 
dicated in the figure as drawn. In other words, we have 
swallowed the relation presented in the drawn figure 
as being universalisable, without having gone through 
the necessary summary induction. 

We may proceed to draw the corrected figure. 
From this we reach, as before, the two conclusions 




But now we see that 



while 



= AF-BF. 



208 CHAPTER IX 

Euclidian demonstration professes to be based on 
pure reasoning, in such a manner that the figure may 
be drawn quite inaccurately, and yet the force of the 
proof be equally cogent. But it may happen, as in the 
case before us, that the figure is drawn with a degree 
of inaccuracy which affects the proof; because the 
particular demonstration, involving unconscious refer 
ence to the figure drawn, has been illegitimately uni- 
versalised. 

7. My explanation of the logical function of the 
figure in geometrical demonstration differs fundamen 
tally from that put forward by Mill, who maintains that 
it is by parity of reasoning that what is apprehended 
to be true for the one drawn figure, is apprehended to 
be true for any other figure (within the scope of the 
conclusion). But the passage from the demonstration 
for one case to that for any other case can only be said 
to exhibit * parity of reasoning when the two demon 
strations have the sameyfr?m Taking for example the 
two demonstrations : 

(1) every m is/; this S is m\ therefore this 5 is/; 

(2) every m\sp\ that 5 is m ; therefore that S is/; 

we may certainly pass by parity of reasoning from (i) 
to (2) inasmuch as both arguments are of the same 
form, the words this and that indicating difference 
in matter. In ascribing the same form to (i) and (2), 
what is meant is that the relation of implication between 
the premisses and conclusion of the one is the same as 
that between the premisses and conclusion of the other. 
But in order to use an implication for the purposes of 
inference, we should have to assert This S is m for 



SUMMARY INDUCTION 209 

case (i) and That S is m for case (2); for although 
the relation of implication is the same in the two argu 
ments, it does not follow from having asserted the 
minor of the one that we can, on this ground, assert 
the minor of the other. Now, in order to establish 
the required conclusion Every S is / we must first 
establish the universalised minor Every S is m. No 
reasoning process (in the accepted meaning of the 
term) would enable us to pass from the case This 5 is 
m to That S is m and to That other 5" is m ad 
infinitum ; and the only mode of establishing the re 
quired universal minor Every 6" is m is through some 
process of induction, the nature of which we have been 
describing. 



J. L. II 



CHAPTER X 

DEMONSTRATIVE INDUCTION 

i. HAVING so far examined intuitive and summary 
induction, we now pass to the third type of inductive 
inference distinguished at the outset, namely demon 
strative induction. As its name suggests, this form of 
inference partakes both of the nature of demonstration 
and of induction. It includes several different forms, 
the characteristics common to them all being (i) that 
they are demonstrative, in the sense that the conclusion 
follows necessarily from the premisses; and (2) that 
they are inductive, in the sense that the conclusion is 
a generalisation of a certain premiss or set of premisses 
which, taken as a collective whole, may be spoken of 
as the instantial premiss/ The possibility of arriving 
demonstratively at a conclusion wider than the premisses, 
depends here upon the nature of the major premiss, 
which is not only universal but composite. In short 
demonstrative induction may be described as that form 
of inference in which one premiss is composite and the 
other instantial; the conclusion being a specification of 
the former and a generalisation of the latter. 

2. In explaining the nature of demonstrative in 
duction as above described, the composite nature of the 
major premiss brings us back to those fundamental 
modes of inference specified in Part I, Chapter III on 
compound propositions. There P and Q are taken to 
stand for any propositions, and four composite relations 



DEMONSTRATIVE INDUCTION 211 

are distinguished in which P may stand to Q\ (a) Im- 
plicative, leading to the Ponendo Ponens\ (b] Counter- 
implicative, leading to the Tollendo Tollens\ (c) Alter 
native, leading to the Tollendo Ponens\ (d) Disjunctive, 
leading to the Ponendo Tollens\ 

(a) UP then Q, but P\ therefore Q. 

(b) If Q then P, but not P] therefore not Q. 

(c) Either P or Q, but not P\ therefore Q. 

(d) Not both P and Q, but/ 5 ; therefore not Q. 

In these composite premisses, we shall take the impli 
cates and alternants to stand for universal propositions, 
and the implicants and disjuncts to stand for particular 
propositions. This secures, for each case, a form of 
inference in which a particular or singular premiss yields 
a universal conclusion. Thus: 

(a) If some 6* is/, then every 7" is q; 

but this S is/, 
.*. every T is q. 

(b} If some T is q, then every S is/; 

but this 6 is/ , 
.*. no T is q. 

(c] Either every S is/, or every T is q\ 

but this 5 is/ , 
. . every T is q. 

(d) It cannot be that some S is/, and some Tisq\ 

but this S is/, 
. . no T is q. 

In the above formulae, it will be observed that the 
simple or categorical premiss is not the precise equiva 
lent or contradictory, as the case may be, of the corre 
sponding proposition that occurs in the composite 
premiss; for this 5 is / is more determinate than 

142 



212 CHAPTER X 

some 5 is /, being one of its superimplicants ; and 
again this 5 is/ is not the mere contradictory of every 
5 is/, being one of its contraries or superopponents. 
The categorical premiss having been in this way 
strengthened, the conditions of valid inference are still 
satisfied. In short, we have taken as our instantial 
premiss a specific instance characterised determinately. 
The object of this is to illustrate the symbolic formulae 
by concrete examples which, when further developed, 
will exhibit the nature of demonstrative induction in its 
most important scientific forms. Consider the following 
illustrations of the symbolic formulae : 

(a) If some one recorded miracle has been shown 
to have happened, then every natural phenomenon has 
a supernatural factor; but such or such recorded miracle 
has been shown to have happened ; therefore every 
natural phenomenon has a supernatural factor. 

(6) If some one female member of a Board had 
lowered the educational standard in her university, 
every woman would have submitted to exclusion from 
the Cambridge Senate; but Miss C. has not submitted 
to exclusion from the Cambridge Senate; therefore no 
female member of a Board has lowered the educational 
standard in her university. 

(c) Either every Protectionist country is financially 
handicapped or every economist of the old school is 
mistaken; but America is commercially prosperous; 
therefore every economist of the old school was mis 
taken. 

(d) It cannot be that some variations can be arti 
ficially produced in domesticated animals, while there 
are some species whose characters are unaffected by 
their environment; but some variations have been arti 
ficially produced in the pigeon; therefore there are no 



DEMONSTRATIVE INDUCTION 213 

species whose characters are unaffected by their environ 
ment. 

These illustrations would be regarded by those logicians 
who divide all inferences into inductive and deductive, 
as being of the nature of deduction rather than of in 
duction, because the universal conclusion is not a 
generalisation of the instantial premiss. In contrast to 
these we will therefore now select a set which will be 
recognised as of the nature of induction; inasmuch as 
here the universal conclusion in each case is a generali 
sation of the instantial premiss. These new examples 
are applications of the same symbolic formulae as the 
preceding set; they differ only in that the symbols S 
and T will now stand for the same class, whereas in the 
first set they stood for different classes. 

(a) If some boy in the school sends up a good 
answer, then all the boys will have been well taught ; 
the boy Smith has sent up a good answer; therefore 
all the boys have been well taught. 

(b) If a single authoritative person had witnessed 
the alleged occurrence, then everyone would have be 
lieved it ; but Mr S. is incredulous ; therefore no 
authoritative person could have witnessed the occur 
rence. 

(c) Either every act of volition is determined or 
every act of volition is free; but by introspection I am 
sure that a certain act of mine was undetermined ; there 
fore every volition is free. 

(ct) It is impossible to suppose that any modern 
theologians are genuine scholars while others have 
remained orthodox; Dean I. is a genuine scholar; 
therefore no modern theologian could have remained 
orthodox. 



214 CHAPTER X 

3. Returning to the symbolically expressed formu 
lae, and substituting p or /, as the case may be, for q, 
as well as S for 7", the composite premisses will assume 
the following still more specialised form: 

(a) If some is p then every ,S is/. 

(6) If some 5* is/ then every S is/. 

(c) Either every 5" is/ or every 6" is/. 

(d) Not both some 5 is/ and some 5 is/. 

It will be seen that these four composite premisses are 
formally equivalent to one another, and that by adding 
the categorical premiss This 5 is / we may conclude 
in each case that Every 5" is/. Now we may transform 
the alternation of universals in (c) and the disjunction 
of particulars in (d) by substituting for / and / any set 
of predicates p, q y r> t, v ... for the alternative pro 
position (c), and the same set in pairs for the disjunctive 
proposition (d), thus : 

(c) Either every 5 is/ or every 5 is q or every S 
is r or ... etc. 

(d) Not both some S is p and some S is q and 
not both some v? is/ and some 5* is r ... etc. etc. 

In this transformation the two complexes (c] and (d) 
are no longer equivalents but rather complementaries 
to one another. If the categorical premiss This S is 
/ is now introduced we may infer by means of (d) that 
No 5 is qj No S is r, No S is f etc., so that all but 
the first of the universal alternants in (c) is rejected, 
and again the universal conclusion Every 5 is / is 
established. The need of combining the complemen 
taries (c) and (d} in order to establish the required uni 
versal conclusion is apparent when we consider a con- 



DEMONSTRATIVE INDUCTION 215 

crete illustration. In the following example, where the 
predicates/, q, r ... stand respectively for attacking 
the Coalitionists/ attacking the Liberals/ attacking 
the Labour Party . . . it will be observed that the com 
posites (c) and (d} retain the same logical force as in 
the above symbolisation, although somewhat differently 
worded : 

(c) At least one of the political parties was attacked 
by every speaker at a certain sitting of the Congress, 
and (d) not more than one of the parties was attacked 
at that sitting. 

Mr X. who spoke attacked the Coalitionist Party. 

. . Every speaker at the sitting attacked the 
Coalitionist Party. 

4. Now, if instead of/, q, r ... we take deter- 
minazes /, / , p" ... under the same determinate P, 
then the disjunctive premiss (d} will not be explicitly 
required, because it is accepted a priori that nothing 
can be characterised by both of any two determinates 
under the same determinate. What remains then is 
the universal alternative proposition (c), established, 
we may assume, by problematic induction ; namely : 
Either every 5 is/, or every Sisp , or every Sisp"... 
running through all the determinates under /Y and 
this may be summed up in the single phrase Every 5 
is characterised by some the same determinate under the 
determinable /Y If to this composite premiss is added 
the mstantial premiss This 5 is pj the universal con 
clusion follows that Every 5 is /. This trio of 
propositions represents the one immediate way of es 
tablishing a generalisation demonstratively from a single 
instance, and it will be termed 



216 CHAPTER X 

The Formula of Direct Universalisation 

Composite Premiss: Every S is characterised by 
some the same determinate under the determinable P. 
Instantial Premiss: This 5 is/. 
Conclusion: . *. Every S is/. 

5. To take a typical illustration from science: 

Every specimen of argon has some the same atcmic 
weight. 

This specimen of argon has atomic weight 39*0. 
.*. Every specimen of argon has atomic weight 39*9. 

In this, as in all such cases of scientific demonstra 
tion, the major premiss is established not direct ty, by 
mere enumeration of instances but rather by deductive 
application of a wider generalisation which has been 
ultimately so established. In the given example it is 
assumed that all the chemical properties of a substance, 
defined by certain test properties, will be the sane for 
all specimens ; and this general formula is applied here 
to the specific substance argon, and to the specific pro 
perty atomic weight. The assumption in this case is 
established by problematic induction, i.e. directly from 
an accumulation of instances. In practically all eiperi- 
mental work, a single instance is sufficient to establish 
a universal proposition : when instances are multiplied 
it is for the purpose of eliminating errors of measure 
ment. It is owing to the fact that the general propo 
sition, functioning as major or supreme premiss, has the 
special form of an alternation of universals thai, by 
means of a minor premiss expressing the result of a 
single observation, we are enabled to establish a uni 
versal conclusion. This conclusion, in accordance with 



DEMONSTRATIVE INDUCTION 217 

our general account of demonstrative induction, is a 
specification of what is predicated indeterminately in 
the universal premiss, and a generalisation of the pro 
position recording the result of a single observed 
instance. 

6. The most important extension of demonstrative 
induction deals with such methods as those of agree 
ment and difference that have been treated by Mill. 
We propose to give a formal account of methods similar 
to those explained by Mill, but so constructed as to 
render them strictly demonstrative. Many critics of 
Mill s methods have treated them disparagingly because 
of his failure to exhibit their formal cogency; while 
others have maintained that induction should not profess 
to exhibit the strictly formal character that is ascribed 
to syllogism and other deductive processes. I hold, on 
the contrary, that Mill s methods can and should be 
exhibited as strictly formal, by rendering explicit certain 
implicit premisses upon which the cogency of the argu 
ment from instances in any given case depends; and 
by indicating the precise conclusion which can be drawn 
from the instances in question. The implicit premiss is 
ultimately established by a process of problematic in 
duction, which must be sharply distinguished from the 
demonstrative process exemplified by the methods. 
Mill s exposition differs from mine, then, in three pre 
liminary respects. In the first place, he does not clearly 
distinguish the nature of direct or problematic induction 
from the nature of the process conducted in accordance 
with his methods of induction/ which he appears often 
to regard as demonstrative. This confusion is particu 
larly noticeable when we contrast his different modes 



218 CHAPTER X 

of treating the methods of Agreement and of Difference : 
Agreement he hardly distinguishes from the method 
of simple enumeration, which is admittedly problematic ; 
whereas Difference he attempts to exhibit as strictly 
demonstrative. In the second place, he professes to 
employ as the supreme major premiss for his methods 
a very wide but at the same time undefined proposition 
called the Law of Causation. In opposition to this 
prevalent view, I hold that it is impossible to present 
such methods as those of Agreement and Difference as 
strictly formal so long as we attempt to subsume them 
under so vague a proposition as the Law of Causation, 
and that each inference drawn in accordance with these 
methods requires its own specific major premiss. The 
formulation of such a major premiss is the necessary 
first step in rendering formally cogent any inference 
(drawn under methods similar to Mill s) from instances 
finite in number, presented either in passive observa 
tion or under experimental conditions. In the third 
place, whereas Mill retains or eliminates a determining 
factor according as it affects or does not affect a deter 
mined character, in my view the precise conclusion to 
be drawn is not correctly expressed in terms of the 
presence or absence of factors, but rather in terms of 
co-variation, thus : according as in two instances a single 
variation in any determining character does or does not 
yield a variation in the determined character, the same 
will hold for any and every further variation of that 
determining character. 

7. In order to obtain the requisite premisses for 
demonstrative induction, we must assume that by a 
preliminary inductive process based upon general ex- 



DEMONSTRATIVE INDUCTION 219 

perience, a number of variable circumstances have been 
eliminated as irrelevant to the formula to be proved. 
The exposition of this preliminary process by which 
irrelevant conditions are eliminated, must be postponed 
until we examine in detail the nature of problematic 
induction. The process itself must be regarded as pre- 
scientific; and science takes up the problem at the 
point where the character of a phenomenon is known to 
depend only upon a limited number of variable con 
ditions. This knowledge is expressed in a proposition 
which constitutes the major premiss in the scientific 
process which we are about to examine as a species of 
demonstrative induction. The major in question is 
specifically different for different classes of phenomena, 
and is in this respect unlike the so-called Law of Causa 
tion which professes to be the same for every class of 
phenomena. If the symbols A, B, C, D, E are taken 
to illustrate the determining characters, and P the 
thereby determined character, then the instances col 
lected in order to establish a given generalisation of 
the form ABCDE ~ P, must be characterised by the 
same set of determinables, and will be said to be of the 
same type or homogeneous with one another. The 
specific major premiss may then be expressed in the 
formula : 

The variations of the phenomenal character P 
depend only upon variations in the characters A, B, C, 
D, E (say). 

The conception of dependence, which the above 
formula introduces, requires more precise explanation. 
In the first place the formula must be understood to 
imply that the variations of A, B, C, D, E, upon which 



220 CHAPTER X 

variations of P depend, are independent of one another. 
For if, for example, a variation of A entailed a varia 
tion of -By then B being a determined character should 
be omitted from amongst the determining characters. 
It is only by observing this principle that we can apply 
the essential rule for all experimentation that one only 
of the determining characters should be varied at a 
time. Again it is essential that A, B, C, D, E, should 
be simplex characters : for the nature of the dependence 
of P upon them is such that, if only one of these 
mutually independent determining characters varies, 
the character P will vary; whereas, if more than one 
of them varied, P might remain constant. This con 
sideration shows that if any character such as A was 
not simplex, but resolvable into unknown factors X 
and Y which varied independently of one another, then 
a variation in A might involve such a variation in 
both X and Y that the character P would remain 
unchanged. 

In the second place, the force of the term * only 
indicates that the dependence of P upon A,B, C, D, E 
is such that no variable circumstances other than these 
need be taken into consideration, all others having 
been previously eliminated in what we have called the 
prescientific or problematic stage of the induction. The 
conclusion that results from this prescientific induction 
is to be expressed by an alternation of universals in 
the form : Either every instance of abcde is/, or every 
instance of abcde is p f , or every instance of abcde is 
/", or.... From this it follows that when a single 
instance is given of abcde that is /, this may be im 
mediately universalised in the form Every abcde is p. 



DEMONSTRATIVE INDUCTION 221 

It should be pointed out that this immediate univer- 
salisation is not dependent upon any comparison of one 
instance with another, and is prior to the use of such 
methods as those of difference or agreement; being in 
fact exemplified above for the case of the atomic weight 
of argon. 

The full significance of the notion of dependence 
is brought out by taking not only instances which agree 
in the determining characters and therefore in the 
determined character, but by taking also instances 
which differ in the determining, and consequently also 
in the determined characters. If a variation in anyone 
of the characters A, B, C, D, E entails a variation in 
P, then, in accordance with the principle underlying 
Mill s method of Difference, that character cannot be 
eliminated ; whereas, if no variation in P is entailed by 
a variation in some one of the characters A.B^C, D, E, 
then, in accordance with the principle underlying Mill s 
method of Agreement, that character can be eliminated. 

8. The forms of Demonstrative Induction to be 
now exhibited contain (i) the supreme premiss of 
dependence formulated above for a given set of deter- 
minables, and (2) a finite set of instantial premisses 
under the same determinables. These forms will be 
distinguished under four heads to be designated figures 
rather than methods ; but will not correspond severally 
to Mill s methods, although primarily based upon his 
method of Difference, and with some important modi 
fications upon his method of Agreement. The notion 
of figure is substituted for that of method ; (a) be 
cause there is only one method employed in the four 
figures, namely that of varying one determining factor 



222 CHAPTER X 

at a time ; and (b) because, as in the case of the figures 
of syllogism, the precise conclusion drawn from the 
instantial premisses will depend on the nature of the 
instances themselves, and the figure to be employed in 
any given case will not be foreknown until the instances 
have been examined and compared. I shall adopt the 
phrases Difference and Agreement for the first two 
figures but Composition and Resolution for the two 
remaining figures. All the four figures have the same 
demonstrative force, and the two last figures though 
they have some resemblance to Mill s or rather 
Herschel s method of Residues, which, as shown in a 
previous chapter, is purely deductive have precisely 
the same inductive .nature as those of Difference and 
Agreement. In each figure, the first step in the demon 
strative process is to universalise each single instance 
taken separately in accordance with the principle of 
Direct Universalisation enunciated above ; and the 
second to draw the more specific conclusion that can 
be inferred from a comparison of instances. 

We proceed to give an account of each of the four 
figures in turn. 

9. Figure of Difference 

Given the supreme premiss : P depends only upon 
ABCDE\ we shall suppose instantial premisses in 
which variations occur in the determining factor D, 
which is assumed to be simplex. 

Then a single instance of abode that is / is uni- 
versalised into Every instance of abcde is/. 

Again a single instance of abcd e that is p r is uni- 
versalised into Every instance of abcd e is/ . 



DEMONSTRATIVE INDUCTION 223 

Comparing these two instances of abce, we note 
that a variation from d to d entails a variation from / 
to/. 

From this we infer that the value of D is actually 
operative in determining the value of P. Hence any 
further variation of D say from d to d" will entail a 
further variation of P say from / to p" . I.e. any value 
of D other than d or d will yield a value of P other 
than/ or/ . 

Represented symbolically, the conclusion reached 
is that 

Every instance of abcd"e will be/ Y 

where this universal is interpreted to signify that, within 
the range abce, any given difference in D will entail 
some difference in P, without however indicating what 
determinate value of P will be yielded by the given 
determinate value of D. 

We may symbolise the form of inference which has 
just been explained in the following scheme : 

Figure of Difference 

Supreme Premiss : P depends only onA,B, C, D, E 
where D is simplex. 

Instantial Premisses Immediate Conclusions 

1. A certain abode is/. . . i. Every abode is/. 

2. A certain abcd e is/ . . .2. Every abcd e is/ . 

Final Conclusion : . . Every abcd"e is/". 

10. Figure of Agreement 

Given the supreme premiss : P depends only upon 
ABCDE-. we shall suppose instantial premisses in 
which variations occur in the determining factor A, 
which is assumed to be simplex. 



224 CHAPTER X 

Then a single instance of abcde that is p is uni- 
versalised into Every instance of abcde is/. 

Again a single instance of a 1 bcde that is / is univer- 
salised into Every instance of a bcde is/. 

Comparing these two instances of bcde, we note 
that a variation from a to a! entails no variation in P. 

From this we infer that the value of A is not 
actually operative in determining the value of P. Hence 
any further variation of A say from a to a" will en 
tail no variation in P\ i.e. any value of A will yield the 
same value/ of P. 

Represented symbolically, the conclusion reached is 

that: 

Every instance of Abcde will yield/, 

where this universal is interpreted to signify that within 
the range bcde, whatever value A may have, the value 
/ will remain unaffected. 

We may symbolise the form of inference which has 
just been explained in the following scheme : 

Figure of Agreement 

Supreme Premiss : P depends only onA,B, C, D, E, 
where A is simplex. 

Instantial Premisses Immediate Conclusions 

1. A certain abcde is p. . . I. Every abcde is p. 

2. A certain a bcde is p. . .2. HLvzrydbcdeisp. 

Final Conclusion : . . Every a" bcde is p. 

11. Figure of Composition 

Given the supreme premiss : P depends only upon 
ABCDE : we shall suppose instantial premisses in 
which variations occur in the determining factor C, 
which is assumed to be simplex. 



DEMONSTRATIVE INDUCTION 225 

Then a single instance of abcde that is / is uni- 
versalised into Every instance of abcde is/. 

Again a single instance of abc de that is/ is univer- 
salised into Every instance of abc de is/ . 

Comparing these two instances of abde we could 
infer, as in the Figure of Difference, that a further 
variation of C would entail a variation in /. But we 
have to contemplate a third instance where c" yields 
the same value / that was presented in the first instance. 
If the values abe are known to be the same as in this 
first instance, then a difference in the remaining factor 
d must have accounted for the recurrence of the 
same determined value /. Thus the first and third 
instances of abe determining / must have been due to 
the compounding of c with d in the first case, and to 
the compounding of c" with d n in the third case. Such 
a case arises when the factor D in the third instance 
has not been amenable to precise evaluation. 

Represented symbolically the conclusion reached 
is that : 

Any instance of abc"pe will be d" } 

where d" is some unevaluated value of D other than 
d or d r . 

Symbolically: 

Figure of Composition 

Supreme Premiss: P depends only onA,B, C, D, , 
where C is simplex. 

Instantial Premisses Immediate Conclusions 

1. A certain abcde is p. .*. i. Every abcde \s p. 

2. A certain abc de is/. . . 2. Every abc de is/. 

Final Conclusion : . . Every abc"pe is d". 

J.L.II 15 



226 CHAPTER X 

12. Figure of Resolution 

Given the supreme premiss : P depends only upon 
ABCDE : we shall suppose instantial premisses in 
which variations occur in the determining factor E, 
which is not here assumed to be simplex. 

Then the three single instances of 

abcde ^p, abode ~p f , abcde" ^/, 
may be respectively universalised into 
Every abcde is/, Every abode* is/ , Every abcde" is/. 

Comparing the first and third of these instances, 
where under the range abed, e and also e" yield /, we 
conclude that E is complex, being resolvable say into 
the two independent factors X, Y \ so that (say) e=xy, 
and *"=*"/ . 

Represented symbolically, the conclusion reached 
is that 

* Every abcdxy is/, and Every abcdx y is/, 

where xy and x"^/ f represent the resolution of e and e n 
to account for the same value/ of P. Thus : 

Figure of Resolution 
Supreme Premiss : / depends only upon A, B, C, D, E. 

Instantial Premisses Immediate Conclusions 

1. A certain abcde is/. i. Every abcde is/. 

2. A certain abcde is/ . 2. Every abcde! is/ . 

3. A certain abcde" is/. 3. Every abcde" is/. 

Final Conclusion : E is resolvable into X Y, 
where e=xy, and J = 



13. It will be seen that each of these figures of 
inductive implication is formally equivalent to a single 



DEMONSTRATIVE INDUCTION 227 

disjunction of four propositions. This fourfold disjunc 
tion may be called : 

The Antilogism of Demonstrative Induction 

Given three instances of the same type exhibiting 
three different values of a given determining character, 
then no case can arise in which : 

(1) \h\tgiven determining character is simplex ; 

(2) the values of the other determining characters 
agree throughout the three instances ; 

(3) the value of the determined character differs in 
two of the three instances ; 

(4) the value of the determined character agrees in 
two of the three instances. 

Symbolically expressed, we cannot have 

B simplex ; and 

an instance of a b c d e that is p, 
,, ,, a b c d e that is / , 
,, ,, a b" c d e that is /. 

Expressing this fourfold disjunction in terms of its 
four equivalent implications, we can formulate the 
Figures of Demonstrative Induction thus : 

(not-4) Figure of Difference : If I and 2 and 3 ; then not-4 

(not-3) Figure of Agreement : If i and 2 and 4; then not-3 

(not-2) Figure of Composition : If I and 3 and 4 ; then not-2 

(not-i) Figure of Resolution : If 2 and 3 and 4; then not-i. 

In symbols this becomes : 

Figure of Difference. 

If B is simplex, and we have 

an instance of a b c d e that is /, 
and an instance of a b f c d e that is / , 
then every instance of a b" c d e will be p" . 

is 2 



228 CHAPTER X 

Figure of Agreement. 

If B is simplex, and we have 

an instance of a b c d e that is p, 

and an instance of a b f c d e that is p, 

then every instance of a b" c d e will be p. 

Figure of Composition. 

If B is simplex, and we have 

an instance of a b c d e that is /, 
and an instance of a b c d e that is p\ 
then any instance of a b n c p e must be d". 

Figure of Resolution. 

If we have an instance of a b c d e that is p, 
and an instance of a b 1 c d e that is p 1 y 
and an instance of a b" c d e that is p, 
then B is complex. 

14. A simple illustration of the Figure of Dif 
ference is afforded by Guy-Lussac s law which connects 
variations in the pressure /, temperature t, and volume 
v, of a specific gas g. Suppose that in two instances 
without changing g and p, a change of temperature 
from t to f is found to entail a change of volume from 
v to z/. From this it can be inferred under the Figure 
of Difference that, with the same gas at the same 
pressure, any further change of temperature, say from 
t to t" , would entail a further change of volume, say 
v to v". This experiment does not prove that for any 
other gas or for any other pressure, a change of tem 
perature would entail a change of volume ; nor does it 
indicate what determinate value of the volume would 
be entailed by any supposed further change of tempera 
ture. It should be observed that the conditions required 



DEMONSTRATIVE INDUCTION 229 

for the Method of Difference namely precise con 
stancy in all but one of the determining factors is 
much more easily realisable when dealing with the 
same body or substance and varying its alterable states 
than when we pass from one to another body or sub 
stance in one of which a character is present and in 
the other absent. Hence the conditions most favour 
able for the application of the Figure of Difference are 
those in which concomitant variations in the determin 
ing and determined factors are observed. For Mill, on 
the other hand, the so-called Method of Concomitant 
Variations was primarily distinguished from the Method 
of Difference in that the latter was concerned with 
presence and absence, and the former with variations 
in degree. He speaks of this method as the one 
necessarily required when we cannot wholly get rid of 
a phenomenon, and are obliged to be satisfied with 
noting the varying degrees with which it is manifested 
from instance to instance ; as if this method were a 
sort of makeshift which had to be put up with when 
recourse to the Method of Difference was impossible. 
But it is precisely in those cases in which we can vary 
the degree of a phenomenon, and not in those that 
can be described as presence and absence, that we can 
be assured that the rigid conditions required by the 
Method of Difference are fulfilled. Mill in adopting 
this position neglected the consideration of the homo 
geneity in any collection of instances brought together 
for comparison under any method of induction what 
ever. In the conception of concomitant variations is 
included not only quantitative variations or variations 
of degree but also qualitative variations under any 



2 3 o CHAPTER X 

given determinable such as colour or sound. To illus 
trate Concomitant Variations, Mill chose the method 
employed in connecting the varying heights of the 
tides with the variations of the position of the sun and 
moon relatively to the earth ; but he presented the 
matter as if the difference in the cogency of this method 
from that of Difference was due to the distinction 
between presence and absence in the latter and varia 
tions of degree in the former ; whereas it is obvious 
that the real deficiency in this application of the Method 
of Concomitant Variations was due to the special nature 
of the case, which made it impossible to secure, in the 
different instances examined, exact agreement in regard 
to the circumstances not known to be irrelevant : e.g. 
the variations of height of the tides might have de 
pended upon variations in the force or direction of the 
wind, or in the shape of the coast, etc. So far then 
from regarding the Method of Concomitant Variations 
as an inferior substitute for that of Difference, if by 
the former is meant variation in the alterable states or 
relations of some one body or substance, and by dif 
ference is meant comparison of two similar bodies in 
one of which some quality is present and in the other 
absent, we must regard the former method as superior 
to the latter. For example : if we attempt empirically 
to establish a causal connection between the prosperity 
or the reverse of a country and its adoption of free 
trade or protection, it would be impossible to find two 
different countries which agreed in all relevant respects 
with the exception of this difference in industrial 
policy ; and hence a change in which one policy was 
replaced by the other within one and the same country 



DEMONSTRATIVE INDUCTION 231 

would afford incomparably more cogent evidence of 
causation than a comparison of the effects in two 
different countries which must necessarily differ in 
very many respects that could not be assumed to be 
irrelevant. 

1 5. To illustrate the Figure of Agreement we may 
take instances used to establish the law that the rate at 
which a body falls in vacuo to the earth is independent 
of its weight. In these instances we keep unchanged 
all the possibly relevant circumstances, such as distance 
from the earth, absence of air, substance and shape of 
the falling body, and vary only the weight. From two 
instances in which the weight alone differs, we find 
that the time occupied in falling through any given 
distance is unchanged. In this way we use the Figure 
of Agreement which might also be called the Figure 
of Indifference, since it picks out a determining con 
dition which is naturally expected actually to modify 
the effect in question, and yet is shown by a comparison 
of instances to be indifferent as regards the determinate 
value of the effect. An illustration of this kind seems 
not to have occurred to Mill, because in his Method 
of Agreement every circumstance except one differs in 
the several instances ; whereas, in my formulation of 
the corresponding figure, every circumstance except 
one agrees in the several instances. In other words, 
as regards the determining factors my Figures of Dif 
ference and of Agreement require the same condition, 
namely a single difference ; whereas Mill contrasts the 
two by defining the Method of Difference as involving 
a single difference and the Method of Agreement as 
involving a single agreement. In fact Mill attempts 



2 3 2 CHAPTER X 

the elimination en bloc of all the varying circumstances 
which distinguish the different instances in which the 
same effect-value is observed, whereas what is required 
in order to give corresponding form to the two methods 
is that we should eliminate as indifferent or irrelevant 
only one circumstance at a time. 

1 6. Having illustrated the figures of Agreement 
and Difference, I will explain the strict procedure 
of using these figures in dealing with a number of 
cause factors and of effect factors conjoined in a set of 
examined instances. Taking as our original major 
premiss: ABCDE ~ PQRT, i.e. the conjunction of 
the cause factors A, B, C, D, E determines the con 
junction of the effect factors P, Q, R, T: it is to be 
remembered that no cause-factors other than those 
enumerated are determinative of the enumerated effect- 
factors, as also that no effect-factors other than those 
enumerated are dependent upon the enumerated cause- 
factors. We then take in turn one cause-factor and 
another and find instances from which we may conclude, 
in regard to a given effect-factor either, in accordance 
with the Figure of Difference, that the factor that is 
varied is actually operative, or in accordance with the 
Figure of Agreement, that such factor is not actually 
operative : and this procedure is repeated for each of 
the effect-characters in turn. Each pair of instances 
compared in this way will lead to a universal conclusion 
under the Figure of Difference or of Agreement as the 
case may be. This Complete method (as it may be 
called) is by no means identical with Mill s Joint 
Method of Agreement and Difference, the use of which 
he advocates only to compensate for the failure to 



DEMONSTRATIVE INDUCTION 233 

secure variation in a single factor ; in this Complete 
method, on the other hand, one cause-factor alone is 
varied in each pair of compared instances. 

This process symbolically expressed serves as an 
exercise in the application of the principles underlying 
demonstrative induction. For example, take the follow 
ing instances : 

( i ) abode ~ pqrt ; ( 2 ) a bcde ~ pq r*t ; 

(3) alfcde ~p f qrt ; (4) a bc de ~f/ j r>t ; 

(5) abcd e ~p q r t. 

From (i) and (2), we eliminate a and a as irrelevant 
to p and t\ and infer Abode ~ pt. From (i) and (3), 
we eliminate b and b as irrelevant to q and / ; and 
infer aBcde ~ qt. From (2) and (4), we eliminate c and 
c f as irrelevant to r and t ; and infer a bCde ~ r t. On 
the other hand, from the comparison of (i) and (5) we 
infer that d and d* cannot be eliminated as ineffective as 
regards either /, / or q, q f or r, /. Hence, under the 
Figure of Difference, we infer abcd"e ~ p"q"^ f . Since, 
however, in these two instances, the variation of D is 
inoperative on T, we also infer, under the Figure of 
Agreement, abcDe ~ t. We may now combine the 
conclusions Abcde ~ pt and abcd"e ~ p"q"i>", and thus 
infer Abcd"e ~ p" . This conclusion expresses the fact 
that, under unchanged conditions bee, A is inoperative 
while D is operative upon P. It should be observed 
that the conclusion abcd"e ~ p"q"r" is contrary to the 
inference drawn by Mill in his Method of Difference ; 
for, according to his formulation of the Method, a 
difference in a single cause-factor entails a difference 
in a single effect-factor. Other inferences such as 



234 CHAPTER X 

from (i) and (2) that a"bcde ~ q"r" may be left to the 
ingenuity of the reader to discover. 

1 7. Before illustrating the two remaining figures, 
it is desirable to explain how the symbols employed in 
my notation are to be practically applied. When the 
characters of two or more cause-factors are represented 
by such symbols as a,b,c, ... two typical cases may arise ; 
( i ) where a,6,c,... represent determinates under different 
determinables A, B, C... ; (2) where two or more of 
them are determinates under the same determinable. 
In the latter case, supposing the symbol B to represent 
the same determinable character as A, the three factors 
a, b, c would more naturally be symbolised by a, a, c. 
Here the recurrence of the symbol a indicates that 
there are two factors conjoined which are existentially 
different the one from the other, although characteris- 
able under the same adjectival determinable. In order 
to symbolise the two-fold manifestation of cause-factors 
characterised under the same determinable, we might 
use the subscripts i, 2, to represent existential plurality ; 
and thus, instead of writing ab, ab , a b, a b , etc., we 
should write a^ t a^, #/# 2 , tf/tf/, etc. For example, 
the adjectival determinable Force may be represented 
by F, and, when two forces enter together as cause- 
factors in producing a certain effect, the possible varia 
tions in which they may be conjoined may be repre 
sented by//,///,///,////, etc. Or again: taking 
the character of a chemical element to be indicated say 
by its atomic weight, we may use A to represent this 
adjectival determinable ; and the possible variations in 
which two elements are conjoined in producing a com 
pound may be represented by a^a^ a^, a{a z , #/0/, etc. 



DEMONSTRATIVE INDUCTION 235 

Since, however, amongst symbolists, any difference of 
symbol such as a and b, is never understood to prohibit 
identity of meaning, while of course an identity of 
symbol is always understood to prohibit a difference 
of meaning, the notation that I have adopted in my 
schematisation of the Figures may still be retained 
without danger of confusion ; and, in any case, it serves 
to represent the general principles of the Figures, 
although in specific cases the special notation indicated 
above may be preferred. 

1 8. As regards the two remaining figures of 
Composition and Resolution, we must point out their 
differences from the figures of Agreement and Dif 
ference, and explain what is meant by the composition 
of causes as contrasted with the combination of causes. 
Although these figures have been exhibited in a form 
according to which their demonstrative cogency is 
equivalent to that of Difference or of Agreement, they 
palpably differ from these latter in two respects. In 
the first place, the predicate of the universal conclusion 
drawn in the last two figures concerns one of the deter 
mining factors D or E, while that in the first two figures 
concerns the determined factor P. In the second place, 
the last two figures introduce the notion of * composi 
tion and its converse resolution these terms being 
used in a special and technical sense which requires 
explanation. 

19. The notion of composition has been long un 
derstood in mathematical physics, where the resultant of 
two directed forces regarded as components is repre 
sented by the diagonal of the parallelogram whose 
sides represent these components. The principle by 



236 CHAPTER X 

which the mechanical effect of two conjoined forces 
can thus be calculated, was contrasted by logicians and 
philosophers with the principle underlying chemical 
formulae, in which the properties of a compound sub 
stance could not be calculated in terms of those of the 
elements combined in the compound. This led to the 
view that there was a fundamental antithesis between 
mechanism and chemism, the former of which involved 
a * composition, the latter a * combination of cause- 
factors. Mill introduced and explained the phrase 
composition of causes (i.e. of cause-factors) and con 
trasted this with the combination of cause-factors, 
specially characteristic of chemical phenomena, and 
also, in his opinion, of many psychological and socio 
logical phenomena. Mill s explanation is not altogether 
satisfactory. I will therefore attempt my own ex 
planation of the antithesis between composition and 
combination. 

When two cause-factors represented, say, by the 
determinables B and C are such that there are certain 
pairs of values, say be and b c 9 , which jointly determine 
the same value/ of an effect character P\ then, referen- 
tially to P, the conjunction Reconstitutes a composition. 
On the other hand, when there are no pairs of values 
under the determinables B and C, such as be and b c*, 
which jointly determine the same value of P\ then, 
referentially to P t the conjunction BC constitutes a 
combination. What is important to note here is that 
the distinction between composition and combination 
is not absolute ; for certain conjunctions of cause- 
factors may constitute a composition referentially to 
one assigned effect-character, and a combination re- 



DEMONSTRATIVE INDUCTION 237 

ferentially to another. For example, when chemical 
elements are conjoined in producing a compound sub 
stance, it is possible to take the weights of certain 
elements and different weights of other elements so as 
to produce a compound of the same weight ; hence 
referentially to the effect weight, the conjunction of 
chemical elements comes under the principle of com 
position. But as regards the chemical character of the 
elements conjoined, it is impossible so to vary these as 
to produce a compound of the same chemical character 
in two different cases ; for instance, the substance 
having the chemical properties of water can only be 
produced by the combination of hydrogen and oxygen. 

This account of the distinction between composition 
and combination is to be regarded as an indication 
rather than as a definition. Expressed mathematically: 
the conjunction of the factors B and C constitute a 
composition, referentially to the effect P, when there 
is a certain function /"such that/ equals/" (b, c) for any 
and every value b and c of B and C. We might there 
fore replace the terms composition and combination 
respectively by the more suggestive terms functional 
and non-functional conjunction. The method of dis 
covering and establishing such functional relations will 
be treated in the next chapter. But we cannot well 
illustrate the figures of Composition and Resolution 
without first modifying their formulation in view of the 
above explanation of the nature of composition. 

20. In the figure of Composition as symbolically 
formulated, we took two instances agreeing as regards 
the determining factors abde, and a third instance 
agreeing with both as regards abe, but in which the 



238 CHAPTER X 

factor D was unamenable to precise calculation. We 
then supposed that, while in the first two instances the 
differences c and d in the determining factor C yielded 
a corresponding difference / and p* in the determined 
factor P\ yet, in a third instance, c" yielded the unex 
pected effect / equivalent to that yielded in the first. 
The unexpectedness of this result was thus accounted 
for either by our inability, in the third instance, to 
measure the factor D, or by our error in supposing that 
its value was still unchanged. Now, instead of illus 
trating our figure by supposing equivalence as regards 
P in the first and third instance a somewhat artificial 
assumption let us suppose rather that in the third 
instance the effect, say/ 3 , was other than that calculated 
by a foreknown formula in which the value of P would 
be given by /"==/(#, b, d , e). On the assumption that 
the correctness of this formula had been properly 
assured by means of the functional extension of the 
Figure of Difference or of Concomitant Variations, 
we should rightly infer that any instance of abd pg 
would entail d" in place of d, so that the effect / 3 , under 
the constant conditions abe, would be due to the com 
position d d", and not merely to c" . 

In this modified form, the Figure of Composition 
can be illustrated by the irregular motions from p to / 3 
of the planet Uranus, the positions a, b, e, of any other 
planets being effectively unaltered while that of the sun 
had changed from c to d . The motion from d to d" of 
an unknown planet, afterwards called Neptune, con 
joined with that of the sun from c to d accounted for 
the unexpected movement of Uranus from p to/ 3 ; in 
other words, a, b, e being constant, p 9 was the same 



DEMONSTRATIVE INDUCTION 239 

function of </ and d" as / was of c and d\ so that P 
was a function, not of C alone, but of C and D com 
pounded. 

A similar illustration of the Figure of Resolution 
is found in the experiments by which the new chemical 
substance argon was discovered by Sir William Ramsay. 
Here the factor E would represent atmospheric nitro 
gen, and its greater weight as compared with that 
of nitrogen prepared from chemical compounds was 
accounted for by the resolution of the atmospheric 
nitrogen into the two components argon and pure 
nitrogen. It should be pointed out that the resolution 
here employed was not a chemical analysis, for argon 
does not combine with any other element (as far as is 
at present known) and therefore the resolution in 
question was a true instance of the converse of com 
position. 

In regard to the illustration of Composition involving 
the discovery of Neptune, and that of Resolution in 
volving the discovery of argon, the precise measure 
ments finally made reduced the inference to a purely 
deductive form, which assumed the character of the 
method of Residues according to my interpretation of 
this method (see p. 118). 



CHAPTER XI 

THE FUNCTIONAL EXTENSION OF DEMONSTRATIVE 

INDUCTION 

i. IN concluding the treatment of demonstrative 
inference I propose to recapitulate the results that have 
been so far reached, and to bring into focus the dis 
tinctions and connections between the several forms of 
inference, deductive, inductive and problematic. I have 
already examined the general notion of function, and 
shown how it is employed in mathematical and other 
processes of deductive inference ; and it remains to 
exhibit this notion as it enters into inductive inference 
this constituting the specifically new topic to be dis 
cussed in the present chapter. 

Pure induction, by which is to be understood that 
which involves no assumption of universal laws, has 
been shown to be the sole direct and ultimate mode of 
generalising from instances examined and theoretically 
enumerable. This species of induction I have called 
problematic because, in my view, the universal proposi 
tions which it establishes must be regarded, not as 
absolutely certified, but as accepted only with a higher 
or lower degree of probability depending upon the 
collective character of the instances enumerated. The 
possibility of establishing such direct generalisations 
depends upon certain postulates, the discussion of which 
raises one of the most important and difficult problems 
of philosophical logic ; and even then, the probability 



FUNCTIONAL INDUCTION 241 

to be attached to generalisations thus established has 
to be determined by reference to the formal principles 
of probability. But, so far as these generalisations 
enter into the account of demonstration, they function 
as major premisses. Demonstrative induction, then, so 
far resembles deduction in that it requires the conjunc 
tion of two types of premisses : ( i ) the major or supreme 
universal premiss, which expresses the relation of 
dependence between one specified set of variables 
and another ; and (2) the minor or instantial premiss 
which sums up the results of single observations or ex 
periments. The major premiss in this mixed form of 
demonstration is formulated, not as a uniformity per 
vading all nature, but as a specified universal holding 
only for the special class of phenomena to which the 
conclusion refers. 

2. A very general statement of the contrast 
between my exposition and Mill s is conveniently intro 
duced at this point. I have deliberately separated the 
treatment of formal or demonstrative induction from 
that of problematic induction. In the latter, the accu 
mulation of instances is all important ; in the former, 
a precise major premiss, relating to a finite and enu 
merable set of determinables, is required in each step 
of the formal process. These major premisses are 
assumed to have been previously established, with a 
higher or lower degree of probability, on the principles 
of problematic induction. The essence of problematic 
as contrasted with formal induction is expressed in 
three statements : first ; no wide generalisation, such 
as that which asserts the uniformity of nature, is in 
volved ; secondly ; the instances compared are not 

j. L. ii 16 



242 CHAPTER XI 

determinately analysed with respect to the variable 
characters upon which the proposed generalisation 
may depend ; and hence, thirdly, an indefinite multi 
plication of instances is required in order to give any 
appreciable value to the probability of the conclusion. 
It is partly for this reason that Mill s account of the 
Method of Agreement differs so considerably from my 
extremely simple Figure of Agreement; for Mill is 
largely thinking, under the title Agreement, of a direct 
method of establishing empirical generalisations to 
which only an inferior degree of probability can be 
attached. The generalisations thus established by 
problematic induction function as major premisses in 
demonstrative processes in one of two ways : either as 
established with what may be called experiential as 
opposed to rational certitude ; or as put forward hypo- 
thetically, and thus as exhibiting forms of implication 
rather than of inference implication being defined, 
as in Chapter I, to be potential or hypothetical 
inference. 

3. The term hypothesis has been used by logicians 
in so very many senses that, in order to obviate logical 
confusion, it will be well to examine its various usages, 
showing how they have developed from one funda 
mental element. This element will be found to be 
definitely epistemic rather than constitutive, and for 
my own purposes I consequently prefer to use the 
phrase * hypothetically entertained, which has an epi 
stemic significance quite independent of the form or 
content of the proposition so entertained. We may 
take in turn the various meanings of the substantive 
hypothesis or the adjective hypothetical that occur 



FUNCTIONAL INDUCTION 243 

in deductive or inductive logic, in order partly to 
connect and partly to contrast its epistemic with its 
other bearings. In traditional formal logic, propositions 
are called hypothetical which are in fact compounded 
out of two categorical propositions, say / and q. In 
this case, while the adjective hypothetical is traditionally 
used to denote a particular species of compound pro 
position, namely that of the form if / then q ; yet at 
the same time the term hypothesis clings firstly to the 
proposition p because in this form it is not actually 
asserted, and next to the proposition q because it is 
only assertible on condition that / has been asserted. 
Thus the adjective hypothetical is actually attached to 
three quite distinct propositions or forms of proposition : 
the compound if/ then q ; the simple proposition / 
itself, which I call the implicans ; and the simple pro 
position q which I call the implicate. Now in order to 
make a first approximation to justifying this confused 
terminology, we must consider its epistemic aspect, and 
we may say that normally both the implicans separately 
and the implicate separately are entertained hypo- 
thetically, while the compound proposition if/ then q 
is entertained assertorically. Hence, even where the 
term hypothetical is used in its most precise technical 
sense, it is applied to a form of proposition assumed 
to be entertained assertorically, the components alone 
of this assertoric compound being entertained hypo- 
thetically. 

The recognition of this ambiguity in the use of the 
term hypothetical resolves the often disputed problem 
of the relation in general between induction and deduc 
tion. When we are concerned with the purely formal 

16 2 




244 CHAPTER XI 

relation of implication as subsisting between the pre 
misses and conclusion of any argument of the general 
nature of a syllogism, then these premisses need only 
be entertained hypothetically; while, at the same time, 
the relation of implication itself is to be conceived, not 
only as assertorically advanced, but even as having the 
highest degree or kind of assertoric certitude. The 
conclusion of a syllogism thus deduced is usually spoken 
of as demonstrated, i.e. as having demonstrative certi 
tude ; although, taken by itself, any kind or degree of 
certainty attaching to it is wholly dependent upon the 
kind or degree of certitude with which the premisses 
are entertained. Taking full advantage, then, of Mill s 
account of the functions and value of the syllogism, 
we may say that the hypothetical conclusion has been 
hypothetically demonstrated, and can only be asser 
torically demonstrated when we have examined and 
tested the truth of the premisses. Only when the major 
premiss has been inductively established can the con 
clusion be entertained categorically, and even then 
with a degree of probability dependent upon that of 
the major premiss ; and ultimately upon the mode of 
induction by which the major has been established. 

4. The problematic nature of the universal ob 
tained by induction and functioning as major premiss 
in a deductive process has led to a confusion between 
the notions problematic and hypothetical, resulting in 
the use of the term hypothesis for any proposition 
entertained with a degree of probability. Thus, when 
Jevons says that all induction is hypothetical, what he 
means is merely that an inductive conclusion has not 
certainty but probability. Thus any inductive generali- 



FUNCTIONAL INDUCTION 245 

sation is commonly called a hypothesis ; and the term 
when applied to a scientific theory may have three 
alternative meanings : first, it may mean that the pro 
position is unproven; secondly, that the proposition 
has an appreciable degree of probability which renders 
it worth considering ; thirdly, that the proposition has 
no appreciable probability at all, and may even be 
known to be false. Besides the epistemic significance 
revealed in all these three alternative meanings, the 
term hypothesis must also be understood to indicate 
the purpose which an unproven universal, definitely 
formulated, fulfils in calculating deductively the con 
clusions to which it would lead. In fact Jevons, in 
describing induction as hypothetical, uses the term in 
two quite different senses : first, in the formal sense, to 
indicate the provisional or tentative attitude towards a 
universal before we have confirmed it by a process in 
volving deduction ; and, secondly, to represent the 
final attitude towards a universal after it has been 
tested and confirmed with the highest attainable degree 
of probability. With the view indicated in the second 
application of the term hypothesis, I agree ; but, as 
regards the first use of the term, it seems to me that 
we always adopt a tentative attitude towards a proposi 
tion entertained as a proposal, whether it is to be proved 
deductively or inductively; so that the term as applied 
to a proposition to be proved does not represent any 
characteristic peculiar to induction. Now the special 
topic with which this chapter is concerned involves 
both the contrasted ideas of hypothesis : namely, of a 
proposition having a certain degree of probability, and 
of one put forward to be tested by appropriate evi- 



246 CHAPTER XI 

dence. Thus, while the functional formula in deduction 
is assumed to be true and therefore may serve as pre 
miss for deducing an equally assured conclusion, the 
inductive aspect of such a functional formula presents 
the inverse problem ; for we have now to examine by 
what kind of instances, and by what modes of com 
parison, the functional formula itself can be established. 
So far as this process of examination may be said 
to have a special characteristic by which it may be 
distinguished from problematic induction used for 
establishing the wide generalisations of science, its 
peculiarity is that a comparatively small number of 
instances will constitute the sufficient factual basis for 
the establishment of the formula, and that the actual 
procedure of mathematical physics, at least in the 
majority of cases, rightly attaches practical certitude 
to the formula thus inferred. 

5. In order to show how the functional formula 
is established, I must refer to my account of the figures 
of Demonstrative Induction. There the conclusion 
demonstratively drawn does not assign the specific 
value of the effect-character that is to be correlated 
with any given value of a cause-character. In popular 
language, the conclusions drawn would be termed 
qualitative not quantitative ; that is to say, the figures 
establish causal connection without determination of a 
causal law or formula. I n comparing the different figures, 
it is seen that the Figure of Difference, which stands 
first, is a direct expression of the principle of the de 
pendence of change in the effect upon change in the 
cause ; and that the Figure of Agreement or of Indif 
ference is complementary to that of Difference in the 



FUNCTIONAL INDUCTION 247 

same sense as the universal or implicative if not-/ then 
not-? is the complementary of if p then q ; while 
the Figures of Composition and Resolution merely 
carry out the principle of Difference under certain 
more complicated circumstances. There is, therefore, 
one principle common to all the four figures, namely 
that underlying the Figure of Difference the functional 
extension of which will be our principal concern. 

The original formula of Difference may be restated 
in the following canon : When in two instances a dif 
ference in the cause-character D entails a difference in 
the effect-character P, all other cause-characters which 
might contribute to the determination of P being the 
same in the two instances, then we infer that any other 
difference in the cause-character will be correlated with 
some other difference in the effect-character, under the 
continued constancy of the remaining cause-characters. 
Now this canon, which applies to two instances only, 
may be obviously extended to any number of instances 
all of which conform to the figure of Difference: i.e. 
all other cause-factors remaining unchanged, we find a 
series of instances in which D alone varies, and in 
which the determinate values d, d , d", d n , etc., say, 
are associated respectively with /, / , p" , p" , etc. Now, 
as in the simple case of two instances, these observa 
tions do not enable us to assign the specific value of P 
that is to be correlated with any given value of D : 
we can still only infer that any further change in D 
will be associated with some further change in P. The 
required extension of the figure of Difference consists, 
therefore, in the determination of P as a function of D 
which shall hold for all unexamined as well as examined 



248 CHAPTER XI 

instances. A famous example of the determination of 
such a function is that formulated by Kepler who, after 
nineteen guesses, discovered a formula for the plane 
tary movements about the sun which co-ordinated 
the spatio-temporal relations for the cases necessarily 
finite in number that he was able to examine and 
measure. The discovery of this formula involved nothing 
of the nature of inductive inference, but its application 
to all the planetary positions intervening between those 
observed constituted a genuine inductive inference, so 
easy to draw that neither Whewell nor Mill seems to 
have been aware that any such inference was implicitly 
made. 

The canon for the Figure of Composition may be 
reformulated as follows : When in several instances 
variations in the single cause-character C have entailed 
variations in the effect-character P such that, in ac 
cordance with the functional extension of the Figure 
of Difference, P has been shown to be a certain func 
tion of C, then, if some similar instance of a further 
variation of C has entailed a variation of P not satisfying 
this function, we infer that, in this instance, besides 
C some other character, say D, has varied, and hence 
that P depends upon the composition of C with D. 
This simple use of the Figure of Composition does 
not, however, enable us to determine the value of D in 
the particular instance observed. In expanding this 
figure therefore we have to look for further instances 
in which both C and D can be evaluated ; and thus 
construct a formula by which P is represented as a 
function both of C and of D, This method should be 
compared with that of Residues, which I have regarded 



FUNCTIONAL INDUCTION 249 

as purely deductive ; for, in the method of Residues, 
the values of D are determined deductively from the 
known formula / =f(c, d], whereas, in our extension of 
the Figure of Composition, the formula /=/ (c, d] is 
determined inductively from the observed values of D. 
The case of the irregularities in the movements of 
Uranus, instanced in the previous chapter, illustrates 
this type of functional extension. 

6. Now the formula which expresses an effect as 
a function of one or more cause-factors must at least 
satisfy the negative condition that it fits all the examined 
instances as regards the observed values of cause and 
effect. Many logicians, and certainly many experi 
menters in practical branches of science, are finally 
satisfied with this negative criterion. They assert, in 
effect, that provided the formula p=f(d\ where /has 
some specific form, agrees with the values of P and D 
as measured in the examined cases, then it has all the 
guarantee that experimentation requires for its uni- 
versalisation. But the mathematician points out that, 
theoretically speaking, there are an infinity of different 
functions that would exactly fit any finite number of 
cases of covariation. Hence he demands in general a 
much more rigid defence for selecting one formula 
rather than another to represent the universal law. 

In order to escape this threatening annihilation of 
inductive inference, we may indicate two fundamental 
principles upon which the highest attainable degree of 
certainty, which may be called practical or experiential 
certitude, depends. In the first place, reliance is placed 
upon the character of the formula itself, and in par 
ticular on its comparative simplicity; in the second 

16-5 



250 CHAPTER XI 

place, the higher credibility of a proposed formula 
depends upon its analogies with other sufficiently well- 
established formulae in similar classes of phenomena. 
Briefly, the criteria of simplicity and analogy, especially 
when conjoined, confer upon a formula of covariation 
that highest degree of probability which allows us to 
regard the induction, not as merely problematic, but as 
virtually demonstrative. For example, the experiments 
that have been conducted in regard to the covariations 
of temperature, pressure and volume of gases have 
always been treated by physicists as conferring absolute 
demonstrative certitude upon the formulae inferred, 
although they have been actually confirmed from a 
necessarily limited number of observations. 

We may illustrate the notion of simplicity by taking 
the simplest of all possible functions, namely where p 

is proportional to d y or its inverse -,. For example, if 

we have instances in which, weight being the deter 
mined factor, and some quantitatively measurable cause 
D varies so that where we double D we double P, and 
where we treble D we treble P, and so on for fractional 
as well as integral multipliers, we inductively infer that 
P, not merely varies with D, but in mathematical 
language, varies as D. There have been philosophers 
who, in effect, have imagined that, unless a causal 
formula can be expressed by a proportionate relation 
of cause to effect, it must be regarded as a mere em 
pirical rule ; and conversely, as soon as instances are 
found to fit some such simple formula, the generalisa 
tion may be regarded as absolutely certified. A slightly 
less simple kind of formula is exemplified by gravitation 



FUNCTIONAL INDUCTION 251 

where, for a given attracting mass, the acceleration of 
the attracted body varies inversely as the square of 
the distance, being in the direction towards the attract 
ing body. The high probability of this formula is due, 
not only to its relative simplicity, but to its analogy 
with the independently known formula for the intensity 
of radiant light or heat. Moreover the formula in 
question could have been deduced from the assumption 
that radiation operates equally in all spatial directions, 
so that its magnitude upon any part of a spherical sur 
face is inversely proportional to the area of that surface 
and therefore to the square of the distance. In the 
examples thus brought forward, indications are given 
of the kind of reasoning upon which the high proba 
bility attached to any formula that fits the examined 
instances is based. 

7. The criterion of simplicity is not often directly 
applicable ; but, when in a relatively complex conjunc 
tion of circumstances that can be analysed, a formula 
is constructed that could have been deduced from a 
combination of wider and well-established formulae of 
comparative simplicity, then an empirical formula thus 
confirmed acquires problematic value corresponding to 
that of the laws from the combination of which it could 
have been deduced. Both Whewell and Mill have 
taken this kind of criterion as fundamental in their 
theories of induction ; Whewell using the phrase con 
silience of inductions, and Mill having in his earlier 
chapters put forward this deductive confirmation as the 
one principle dominating his whole theory. At first 
sight Mill s position is paradoxical, since he apparently 
attributes a higher probability-value to a law, merely 



252 CHAPTER XI 

on the ground of its width, whereas it would appear 
that the narrower generalisation is the safer. I think, 
on this matter, we must recognise the value of the two 
opposed principles that have been put forward. On 
the one hand, mere simplicity has been elevated into 
a supreme criterion ; but, so far from admitting that 
simplicity alone guarantees a formula, we must main 
tain that where a known complexity of circumstances is 
involved, a corresponding complexity must be expected 
to characterise their co-ordinating formula. Hence, 
when a class of phenomena that have not been defini 
tively analysed resembles other classes for which a 
complex formula has been established, a corresponding 
complexity should be anticipated for the given class ; 
whereas the formula for a class of phenomena analogous 
to others for which a simple formula holds may rightly 
be expected to be simple. The criterion of simplicity, 
when including its indirect as well as its direct form, is 
of value ; but it is only when analogy is thus conjoined 
with simplicity that we may attach practical certitude 
to a formula which satisfies at least the negative cri 
terion of fitting perhaps only a small number of well- 
examined cases. 

8. The theory of what I have called the functional 
extension of demonstrative induction constitutes a link 
between the Demonstrative and the Problematic forms 
of inference. For certain rules (of a strictly formal 
character) are required for deducing, amongst all the 
functions which fit the observed co- variations, the most 
probable function of the variable cause-factors by which 
an effect-factor may be calculated. The oldest and most 
usual method of determining this function is known as 



FUNCTIONAL INDUCTION 253 

the method of least squares. Its validity depends upon 
a certain assumption with regard to the form of the 
Law of Error, i.e. of the function exhibited by diver 
gences from a mean or average, when the number of 
co-variational instances is indefinitely increased ; and a 
different method must be employed for each correspond 
ing different assumption. The reader must be referred 
to Mr J. M. Keynes s Treatise on Probability, Chapter 
XVII, for a very comprehensive and original discussion 
of this topic. 

The inductive inference examined in the above is 
thus shown to be based upon purely formal and demon 
strative principles of probability, whereas the discussion 
of problematic induction to be developed in Part III 
will introduce informal theorems of probability, based 
on postulates of a highly controversial nature. It is 
therefore legitimate, and even necessary, to include the 
functional extension of the figures of induction under 
the general title of demonstrative inference. 



INDEX 



Abstraction 148, 166 ; psychological 
account of 190 

Adjectives, and abstraction 148; 
compound 61, 64; and mathe 
matical concepts 140; nature of 
xiii 

Agreement, figure of 223, 228 ; il 
lustrations of 231 ; Mill s method 
of 118, 217, 242 

Algebra, and functional deduction 
124, 130; and logical principles 

135 

Algebraical dimensions 185 ; proof 
201 

Alphabet and numerical notation 
158 

Alternative relation of propositions 
211 

Analogy, a criterion of certitude 2 50 

"And", conjunctive 63; enumera- 
tive 62 

Antilogism 78 ; for demonst. induc 
tion 227 ; for syllogism 80, 87 

Applicative principle 10, 27, 104, 
1 1 8, 123, 129; in mathematics 
132 

Aristotle s doctrine of proprium 125 

Arithmetic, and logic 133 ; and 
number 158 

Arithmetical processes 181 

Assertion and the proposition xiv,65 

Assertoric and hypothetic 243 

Association and inference 3, 7 

Associative Law 128 

Attention 190 

Axioms, establishment of 33, 201 ; 
geometrical 201 ; of mathema 
tics 123; and necessary inference 
126 



Boole s symbolic logic 136 

Boyle s Law 107, no 

Brackets, function of 53, 122, 129 

Cantor 128, 137, 176 

Carroll, Lewis 77 

Categories, definition of 15; and 
latent form 55, 60, 139; and 
magnitude 154 

Causal formula 246 

Causation, Law of 218 

Cause and effect, and figures of in 
duction 232 ; and absolute mea 
surement 179; reversibility 107,. 
116 

Certitude, criteria of 249; demon 
strative 250; experiential and 
rational 242; of hypothetical pro 
positions 244 ; of intuitive gene 
ralisations 192 

Characterisation, a relational predi 
cation 142 

Classes, "comprising" items 146, 
167 ; and genuine constructs 62 ; 
and extensional wholes 166; and 
number 154; and series 155; 
and syllogism 87 

Class-names and symbolic variables 
60 

Class-terms and syllogism 79, 84 

Combination and composition 236 

Commutative Law 128 

Composite propositions and demon 
strative induction 212 

Composition, and combination 236 ; 
figure of 222, 224, 228; illustra 
tions of 238, 248 ; principle under 
lying 248 

Compounds, nature of 61 



INDEX 



255 



Comprising, and classes 146; a 

relational predication 142 
Conjunctional functions 55, 62, 72 
Connectional functions 54, 57, 141 
Connotation and property 125 
Constants, absolute and relative 
120; formal and material 43, 
141 ; implicit and explicit 53 
Constitutive condition of inference 

8, 10 

Constructs, fictitious 61, 64; and 
functions 48; simple and com 
pound 141 
Continuants xi, 1 10 
Conversion 31, 39; a type of intui 
tion 195 ; relative 100 
Correlation, factual and factitious 
I S^, 159; functional 160; one- 
one 158 

Counter-applicative principle 28 
Counter-implicative principle 29 ; 

relation 211 

Counter-principles of inference 28 
Counting, analysis of act 157; 
logical principles underlying 
158 

Co-variation, in economics 115; 
formulae establishing 249; and 
inductive figures 218, 219, 229; 
law of 106; in physics 113 

Deduction 104; functional 129; 
and observation 119; range of 
189, 213; and method of Resi 
dues 1 1 8 ; employment in Science 
216 

Demonstrative induction 210; cer 
titude of 250; figures of 222, 227 ; 
Mill s methods 217, 222; use in 
Science 216 

Demonstrative inference 33, 102, 
132; and deduction 241; and 
problematic inference 132, 189, 
241 

Dependence, concept of 219 



Determinables, and categories 19 ; 
in demonst. induction 215; and 
determinates 43, 62, 149, 195 ; 
and distensive magnitudes 169; 
and intensive magnitudes 172 

Difference, figure of 222, 227 ; illus 
tration of 228 ; Mill s method of 
118; principle underlying 247 

Disjunctive propositions 211; prin 
ciple, and the syllogism 78 

Distensive magnitudes 162, 168, 

173 
Distribution 89, 198; syllogistic 

rules of 92 
Distributive Law 128 
Division, concrete 183; contrasted 

with addition 181, 188 

Enthymeme 100 

Epistemic condition of inference 8 ; 
nature of term "hypothesis" 242 

Equality, measurement of 178 ; nu 
merical 145, 149, 159 

Equations, connectional 112; func 
tional 126; limiting 127; linear 
107, 117 

Ethical judgments and intuition 
194 

Euclid 201, 204 

Experiential certification 36 

Experimentation, rule for 220; con 
ditions for valid 249 

Extension, applications of term 1 66 ; 
a species of magnitude 166,174 

Factitious correlations 156,158 
Factual and factitious correlation 

156, 159 

Fallacies, material and formal 101 

Fechner s "just perceptible differ 
ence" 170 

Figures of induction 22 1 ; illustra 
tions of 228 ; use of 232 

Figures of syllogism 77, 87; dicta 
for first three 80, 83 ; fourth 87 



256 



INDEX 



Form, of argument 208 ; elements 
of 53; and matter 191; and pri 
mitive ideas 138 

Formal correlation 160; and ma 
terial 139 ; relations, table of 144 

Formulae, establishment of 33, 127, 
129, 195; of functional induction 
249; range of 129, 131 

Functional conjunction 237 ; corre 
lation 160; deduction 124; in 
duction 246; syllogism 103, 106, 
120, 127 

Functions, conjunctional 72 ; con 
nected and disconnected 130; 
and constructs 48, 130; descrip 
tive 69; formal and non-formal 
5> 75; prepositional 71; and 
variants 49, 57; varieties of 55, 
66,68 

Geometrical figures, use of 201, 203 ; 
abuse of 206 

Geometrical induction 197, 205 ; 
magnitudes 187; proof 201, 204 

Geometry, analytical 204 ; and 
functional deduction 124; Mill 
on foundations of 191 

Gravitation, an instance of function 
al syllogism 109; probability of 
formula 250 

Grounds of argument 38 

Hume s philosophy 82 
Hypothetical propositions 11, 242; 
and problematic 244 

Identity, of adjectives 149; rela 
tion of 20, 142 

Illustrations, choice for syllogism 
77, 8 1, 101 ; of demonstrative in 
duction 212, 213, 215, 216; of 
summary induction 197, 198 

Illustrative symbols 41, 46 

Imagery, and geometrical induction 
202 ; and intuited universals 193 



Implication, and demonst. induction 
210; and hypotheses 243; rela 
tion to inference xv, i, 76 ; a rela 
tional predication 142 

I mplicative formula 152; principle 
10, 27, 104, 118; relation 211 

Including, and extensional wholes 
167; a relational predication 142 

Independence, notional and con- 
nectional 108 

Induction, relation to Deduction 
189,213,243; demonstrative 189, 
210, 227; figures of 221; and 
functional formulae 105, 131 ; in 
tuitive 29, 189; mathematical 
J 3 2 > J 33; an d observation 119; 
pre-scientific 219; problematic 
189, 216, 219, 240; type of Pro 
position underlying 66 ; pure 
240; summary or perfect 197 

Inductive principle 23, 38 

Inference, and implication i, 76, 
152; paradox of 10, 136; pre 
requisites of 2 : principles of 10 ; 
psychological conditions of 4; 
conditions for validity 7 

Infinity, and cardinal numbers 161 ; 
orders of 128; transfinite aggre 
gates 155, 1 60 

Instantial premiss 210, 216 

Integers, finite 133, 161 ; notion of 
139, 154; odd and even 161 

Intensity and reality 172 
Intuition, and experience 191 ; in 
inference 31, 33; and sensation 
192; of space 202; and syllogism 
90 

Intuitive induction 29, 189; and 
certitude 192 ; experiential and 
formal 192; and logical formulae 
195 ; involved in geometry 205 ; 
distinguished from summary 200 

Jevons, Elementary Lessons 116, 
125 ; on induction 244 



INDEX 



257 



Kant s views on geometry 202; 
philosophy 82 

Keynes, J. M., Treatise on Proba 
bility 253 

Language and symbolism 44 

Laws of Nature 106, 126 

Logic, relation to mathematics 
123, 132, 137, 141 ; relation to 
science 216, 228, 231, 235; sym 
bolic 136 

Magnitudes, absolute and relative 
205; abstract and concrete 161, 
181 ; comparison of 174; disten- 
sive 1 68; etymology of 153; ex 
tensive 162; intensive 172; and 
material variables 144; simple 
and compound 180; varieties of 
150, 162, 187 

Major term 76; rules for 94 

Mathematical induction 133; sym 
bolism 136, 141 

Mathematics, and functional formu 
lae 105,112,120,126; and relation 
to logic 123, 137, 141, 151; and 
principles of inference 132, 152 

Measurement, of extensive magni 
tudes 175; of geometrical mag 
nitudes 187 

Middle term 77 ; rules for 93 

Mill, J. S., on foundations of geo 
metry 191, 208 ; inductive methods 
217, 229, 332 ; inductive methods 
criticized 217,233,241 ; on perfect 
induction 197 ; on probability 
value 251; definition of "pro- 
prium" 125; method of Residues 
116, 118, 222; on syllogism xvii, 
244 

Minor term 76 ; rules for 94 

Mnemonic verses 97 

Moods of syllogism 76, 84 ; rules 
for valid 86 

Multiplication, concrete 181; con 
trasted with addition 181, 188 



Number, alphabetical notation of 
158; cardinal and ordinal 155, 
161; and classes 154; psycho 
logical aspect of 155 

Obversion 91, 99 

Occurrents xi 

Operators, logical status of 141 ; 
and number 158 

"Or," function in genuine con 
structs 63 

Order, serial and temporal 157 

Particulars and universals 191, 192 

Peano 137 

Per, meaning of 183 

Perception, analysis of 190; and 
inference 5 

Petitio principii xvii, 10, 136 

Postulates, of problematic induction 
189, 240; of science 219 

Predesignations and functions 69 

Predicational functions 56, 72 

Premisses, composite 210; in in 
ductive figures 218; instantial 
210, 216; subminor and super- 
major 21; of syllogism 76 

Principia Mathematica 66, 138 

Principles, enumeration of 32 ; epi- 
stemic character of 31 ; function 
of 23; of inference 10; underly 
ing inductive figures 247, 248; 
underlying mathematics 123, 158 

Principles of Mathematics xiii, 155, 
161, 165 

Probability, conditions for high de 
gree of 251 ; law of error 253 

Problematic induction, and func 
tional 246; and prescientific in 
vestigation 216, 219, 220, 240 

Problematic inference, and demon 
strative 132, 189,218; and hypo 
thetical 244; and summary in 
duction 198, 200 

Proof, analytical and geometrical 
201 ; science of 200 



INDEX 



Proper names and numbers 156 
Property, notion of 125 
Prepositional functions 66, 71 ; 

types 66 
Propositions, and assertion xiv; 

composite 210; structural 14 
Psychological account of inference 

4 ; account of symbolism 44 

Quantity, relation to magnitude 162 

Ratios, and addenda 171 ; and an 
gles 1 86; notion of 139 

Relational predications 142; many- 
one 145; many-many 156; one- 
one 158 

Relations, adjectival nature of xii ; 
extensional treatment of xii, 1 59 

Residues, Herschel s method of 
1 1 8, 222, 249; Mill s method of 
116 

Resolution, figure of 222, 226, 228 ; 
illustration of 239 

Reversibility, principle of 107, 116 

Russell B., principle of abstraction 
146; notion of class 148; on 
equality 146, 159, 175; notion of 
function 52, 66; on symbolism 
138; on time and space 165; 
theory of types 73 

Science, and demonst. induction 
216; and inductive figures 228, 
23 1 ) 2 35 ; postulate of 219 

Sensational magnitude 170, 180 

Sense-data and induction 38 

Sense-experience, and intuition 192 ; 
nature of 191 

Sentence and proposition 59 

Simple enumeration 218 

Simplicity, a criterion of certitude 
250 

Sorites 97 

Space, Euclidian and non-Euclid 
ian 201 ; measurement of 176; 
relativity of 165 



Stretches, quantitative measure 
ment of 178; varieties of 163 
Structural propositions 14 
Substantive, compound 61 ; nature 

of xi 

Subsumption 103, 120, 124 
Summary induction 200 
Supernumerary moods 85, 88 
Syllogism, analysis of 12, 17, 76; 
dicta for figures 80, 83 ; functional 
103, 1 20, 127; illustrations of 77, 
81, 101 ; importance of 102 ; and 
mathematics 123; Mill s analysis 
xvii; principle of 21, 24; and 
summary induction 197; and 
thought process 160; rules for 
valid moods 89 

Symbolism, use in inductive figures 
234; mathematical 130, 141 ; and 
meaning 45 ; psychological ac 
count of 44 ; value of 39, 41, 136 ; 
varieties of 41, 129 

Ties, nature of 53; temporal and 
spatial 164 

Time and space, logical nature of 
163; measurement of 176; rela 
tivity of 165 

Universal propositions 1 1 
Universalisation, formula of 216, 

220, 222 
Universals, apprehension of 191 

Variables, apparent 58, 66 ; in func 
tional formulae 108, 112, 120, 127, 
130; formal and material 140, 144 

Variants 71 

Verbal propositions 125 

Verification 119 

WhewelFs defence of perfect in 
duction 199 

Wholes, and parts 162; extensive 
and extensional 166 ; in geometry 
204 



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4 1973 



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